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%I M3037 N1231 #432 Jun 27 2023 12:04:08
%S 1,3,17,99,577,3363,19601,114243,665857,3880899,22619537,131836323,
%T 768398401,4478554083,26102926097,152139002499,886731088897,
%U 5168247530883,30122754096401,175568277047523,1023286908188737,5964153172084899,34761632124320657
%N a(0) = 1, a(1) = 3; for n > 1, a(n) = 6*a(n-1) - a(n-2).
%C Chebyshev polynomials of the first kind evaluated at 3.
%C This sequence gives the values of x in solutions of the Diophantine equation x^2 - 8*y^2 = 1, the corresponding values of y are in A001109. For n > 0, the ratios a(n)/A001090(n) may be obtained as convergents to sqrt(8): either successive convergents of [3; -6] or odd convergents of [2; 1, 4]. - _Lekraj Beedassy_, Sep 09 2003 [edited by _Jon E. Schoenfield_, May 04 2014]
%C Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r = sqrt(8). - _Benoit Cloitre_, Feb 14 2004
%C Appears to give all solutions greater than 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r = sqrt(2). - _Benoit Cloitre_, Feb 24 2004
%C This sequence give numbers n such that (n-1)*(n+1)/2 is a perfect square. Remark: (i-1)*(i+1)/2 = (i^2-1)/2 = -1 = i^2 with i = sqrt(-1) so i is also in the sequence. - _Pierre CAMI_, Apr 20 2005
%C a(n) is prime for n = {1, 2, 4, 8}. Prime a(n) are {3, 17, 577, 665857}, which belong to A001601(n). a(2k-1) is divisible by a(1) = 3. a(4k-2) is divisible by a(2) = 17. a(8k-4) is divisible by a(4) = 577. a(16k-8) is divisible by a(8) = 665857. - _Alexander Adamchuk_, Nov 24 2006
%C The upper principal convergents to 2^(1/2), beginning with 3/2, 17/12, 99/70, 577/408, comprise a strictly decreasing sequence; essentially, numerators=A001541 and denominators=A001542. - _Clark Kimberling_, Aug 26 2008
%C Also index of sequence A082532 for which A082532(n) = 1. - _Carmine Suriano_, Sep 07 2010
%C Numbers n such that sigma(n-1) and sigma(n+1) are both odd numbers. - _Juri-Stepan Gerasimov_, Mar 28 2011
%C Also, numbers such that floor(a(n)^2/2) is a square: base 2 analog of A031149, A204502, A204514, A204516, A204518, A204520, A004275, A001075. - _M. F. Hasler_, Jan 15 2012
%C Numbers such that 2n^2 - 2 is a square. Also integer square roots of the expression 2*n^2 + 1, at values of n given by A001542. Also see A228405 regarding 2n^2 -+ 2^k generally for k >= 0. - _Richard R. Forberg_, Aug 20 2013
%C Values of x (or y) in the solutions to x^2 - 6xy + y^2 + 8 = 0. - _Colin Barker_, Feb 04 2014
%C Panda and Ray call the numbers in this sequence the Lucas-balancing numbers C_n (see references and links).
%C Partial sums of X or X+1 of Pythagorean triples (X,X+1,Z). - _Peter M. Chema_, Feb 03 2017
%C a(n)/A001542(n) is the closest rational approximation to sqrt(2) with a numerator not larger than a(n), and 2*A001542(n)/a(n) is the closest rational approximation to sqrt(2) with a denominator not larger than a(n). These rational approximations together with those obtained from the sequences A001653 and A002315 give a complete set of closest rational approximations to sqrt(2) with restricted numerator or denominator. a(n)/A001542(n) > sqrt(2) > 2*A001542(n)/a(n). - _A.H.M. Smeets_, May 28 2017
%C x = a(n), y = A001542(n) are solutions of the Diophantine equation x^2 - 2y^2 = 1 (Pell equation). x = 2*A001542(n), y = a(n) are solutions of the Diophantine equation x^2 - 2y^2 = -2. Both together give the set of fractional approximations for sqrt(2) obtained from limited fractions obtained from continued fraction representation to sqrt(2). - _A.H.M. Smeets_, Jun 22 2017
%C a(n) is the radius of the n-th circle among the sequence of circles generated as follows: Starting with a unit circle centered at the origin, every subsequent circle touches the previous circle as well as the two limbs of hyperbola x^2 - y^2 = 1, and lies in the region y > 0. - _Kaushal Agrawal_, Nov 10 2018
%C All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0<a<b<c are given by a=A001542(n), b=A005319(n), c=A001542(n+1), x=A001541(n), y=A001653(n+1), z=A002315(n) with 0<n. - _Michael Somos_, Jun 26 2022
%D Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)
%D J. W. L. Glaisher, On Eulerian numbers (formulas, residues, end-figures), with the values of the first twenty-seven, Quarterly Journal of Mathematics, vol. 45, 1914, pp. 1-51.
%D G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium 194 (2009), 185-189.
%D A. Patra, G. K. Panda, and T. Khemaratchatakumthorn. "Exact divisibility by powers of the balancing and Lucas-balancing numbers." Fibonacci Quart., 59:1 (2021), 57-64; see C(n).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - _N. J. A. Sloane_, Mar 08 2022
%H T. D. Noe, <a href="/A001541/b001541.txt">Table of n, a(n) for n = 0..200</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), 181-193.
%H Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2006.07566">Lattice Equable Parallelograms</a>, arXiv:2006.07566 [math.NT], 2020.
%H Jean-Paul Allouche, <a href="https://doi.org/10.1051/epjconf/202024401008">Zeta-regularization of arithmetic sequences</a>, EPJ Web of Conferences (2020) Vol. 244, 01008.
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H H. Brocard, <a href="https://gdz.sub.uni-goettingen.de/id/PPN598948236_0004?tify={%22pages%22:[186],%22view%22:%22info%22}">Notes élémentaires sur le problème de Peel</a>, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.
%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H P. Catarino, H. Campos, and P. Vasco, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_45_from11to24.pdf">On some identities for balancing and cobalancing numbers</a>, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.
%H Kwang-Wu Chen and Yu-Ren Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Pan/pan32.html">Greatest Common Divisors of Shifted Horadam Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.5.8.
%H S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.
%H Robert Frontczak, <a href="https://doi.org/10.12988/ams.2018.87111">A Note on Hybrid Convolutions Involving Balancing and Lucas-Balancing Numbers</a>, Applied Mathematical Sciences, Vol. 12, 2018, No. 25, 1201-1208.
%H Robert Frontczak, <a href="http://www.m-hikari.com/ijma/ijma-2018/ijma-9-12-2018/p/frontczakIJMA9-12-2018.pdf">Sums of Balancing and Lucas-Balancing Numbers with Binomial Coefficients</a>, International Journal of Mathematical Analysis (2018) Vol. 12, No. 12, 585-594.
%H Robert Frontczak, <a href="https://doi.org/10.12988/ijma.2019.9211">Powers of Balancing Polynomials and Some Consequences for Fibonacci Sums</a>, International Journal of Mathematical Analysis (2019) Vol. 13, No. 3, 109-115.
%H Robert Frontczak, <a href="http://nntdm.net/papers/nntdm-25/NNTDM-25-2-169-180.pdf">Identities for generalized balancing numbers</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 169-180.
%H Robert Frontczak and Taras Goy, <a href="https://arxiv.org/abs/2007.14048">Additional close links between balancing and Lucas-balancing polynomials</a>, arXiv:2007.14048 [math.NT], 2020.
%H Robert Frontczak and Taras Goy, <a href="https://arxiv.org/abs/2007.14618">More Fibonacci-Bernoulli relations with and without balancing polynomials</a>, arXiv:2007.14618 [math.NT], 2020.
%H Robert Frontczak and Taras Goy, <a href="https://arxiv.org/abs/2009.09409">Lucas-Euler relations using balancing and Lucas-balancing polynomials</a>, arXiv:2009.09409 [math.NT], 2020.
%H O. Khadir, K. Liptai, and L. Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Szalay/szalay11.html">On the Shifted Product of Binary Recurrences</a>, J. Int. Seq. 13 (2010), 10.6.1.
%H J. M. Katri and D. R. Byrkit, <a href="http://www.jstor.org/stable/2313820">Problem E1976</a>, Amer. Math. Monthly, 75 (1968), 683-684.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H D. H. Lehmer, <a href="http://projecteuclid.org/euclid.dmj/1077490638">A cotangent analogue of continued fractions</a>, Duke Math. J., 4 (1935), 323-340.
%H D. H. Lehmer, <a href="/A002065/a002065_1.pdf">A cotangent analogue of continued fractions</a>, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.
%H Dino Lorenzini and Z. Xiang, <a href="http://alpha.math.uga.edu/~lorenz/IntegralPoints.pdf">Integral points on variable separated curves</a>, Preprint 2016.
%H aBa Mbirika, Janeè Schrader, and Jürgen Spilker, <a href="https://arxiv.org/abs/2301.05758">Pell and associated Pell braid sequences as GCDs of sums of k consecutive Pell, balancing, and related numbers</a>, arXiv:2301.05758 [math.NT], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.html">J. Int. Seq.</a> (2023) Vol. 26, Art. 23.6.4.
%H Robert Phillips, <a href="https://web.archive.org/web/20100713033314/http://www.usca.edu/math/~mathdept/bobp/pdf/polgonal.pdf">Polynomials of the form 1+4ke+4ke^2</a>, 2008.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Prasanta K. Ray, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/17-20-2012/kumarrayIJCMS17-20-2012.pdf">Curious congruences for balancing numbers</a>, Int. J. Contemp. Math. Sci. 7 (2012), 881-889.
%H Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=18">Integer sequences and ellipse chains inside a hyperbola</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
%H N. J. Wildberger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Wildberger/wildberger2.html">Pell's equation without irrational numbers</a>, J. Int. Seq. 13 (2010), 10.4.3, Section 4.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F G.f.: (1-3*x)/(1-6*x+x^2). - _Barry E. Williams_ and _Wolfdieter Lang_, May 05 2000
%F E.g.f.: exp(3*x)*cosh(2*sqrt(2)*x). Binomial transform of A084128. - _Paul Barry_, May 16 2003
%F From _N. J. A. Sloane_, May 16 2003: (Start)
%F a(n) = sqrt(8*((A001109(n))^2) + 1).
%F a(n) = T(n, 3), with Chebyshev's T-polynomials A053120. (End)
%F a(n) = ((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2.
%F a(n) = cosh(2*n*arcsinh(1)). - _Herbert Kociemba_, Apr 24 2008
%F a(n) ~ (1/2)*(sqrt(2) + 1)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F For all elements x of the sequence, 2*x^2 - 2 is a square. Limit_{n -> infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - _Gregory V. Richardson_, Oct 10 2002 [corrected by Peter Pein, Mar 09 2009]
%F a(n) = 3*A001109(n) - A001109(n-1), n >= 1. - _Barry E. Williams_ and _Wolfdieter Lang_, May 05 2000
%F For n >= 1, a(n) = A001652(n) - A001652(n-1). - _Charlie Marion_, Jul 01 2003
%F From _Paul Barry_, Sep 18 2003: (Start)
%F a(n) = ((-1+sqrt(2))^n + (1+sqrt(2))^n + (1-sqrt(2))^n + (-1-sqrt(2))^n)/4 (with interpolated zeros).
%F E.g.f.: cosh(x)*cosh(sqrt(2)x) (with interpolated zeros). (End)
%F For n > 0, a(n)^2 + 1 = 2*A001653(n-1)*A001653(n). - _Charlie Marion_, Dec 21 2003
%F a(n)^2 + a(n+1)^2 = 2*(A001653(2*n+1) - A001652(2*n)). - _Charlie Marion_, Mar 17 2003
%F a(n) = Sum_{k >= 0} binomial(2*n, 2*k)*2^k = Sum_{k >= 0} A086645(n, k)*2^k. - _Philippe Deléham_, Feb 29 2004
%F a(n)*A002315(n+k) = A001652(2*n+k) + A001652(k) + 1; for k > 0, a(n+k)*A002315(n) = A001652(2*n+k) - A001652(k-1). - _Charlie Marion_, Mar 17 2003
%F For n > k, a(n)*A001653(k) = A011900(n+k) + A053141(n-k-1). For n <= k, a(n)*A001653(k) = A011900(n+k) + A053141(k-n). - _Charlie Marion_, Oct 18 2004
%F A053141(n+1) + A055997(n+1) = a(n+1) + A001109(n+1). - _Creighton Dement_, Sep 16 2004
%F a(n+1) - A001542(n+1) = A090390(n+1) - A046729(n) = A001653(n); a(n+1) - 4*A079291(n+1) = (-1)^(n+1). Formula generated by the floretion - .5'i + .5'j - .5i' + .5j' - 'ii' + 'jj' - 2'kk' + 'ij' + .5'ik' + 'ji' + .5'jk' + .5'ki' + .5'kj' + e. - _Creighton Dement_, Nov 16 2004
%F a(n) = sqrt( A055997(2*n) ). - _Alexander Adamchuk_, Nov 24 2006
%F a(2n) = A056771(n). a(2*n+1) = 3*A077420(n). - _Alexander Adamchuk_, Feb 01 2007
%F a(n) = (A000129(n)^2)*4 + (-1)^n. - _Vim Wenders_, Mar 28 2007
%F 2*a(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + A001542(k)^2. - _Charlie Marion_, Oct 12 2007
%F a(n) = A001333(2*n). - _Ctibor O. Zizka_, Aug 13 2008
%F A028982(a(n)-1) + 2 = A028982(a(n)+1). - _Juri-Stepan Gerasimov_, Mar 28 2011
%F a(n) = 2*A001108(n) + 1. - _Paul Weisenhorn_, Dec 17 2011
%F a(n) = sqrt(2*x^2 + 1) with x being A001542(n). - _Zak Seidov_, Jan 30 2013
%F a(2n) = 2*a(n)^2 - 1 = a(n)^2 + 2*A001542(n)^2. a(2*n+1) = 1 + 2*A002315(n)^2. - _Steven J. Haker_, Dec 04 2013
%F a(n) = 3*a(n-1) + 4*A001542(n-1); e.g., a(4) = 99 = 3*17 + 4*12. - _Zak Seidov_, Dec 19 2013
%F a(n) = cos(n * arccos(3)) = cosh(n * log(3 + 2*sqrt(2))). - _Daniel Suteu_, Jul 28 2016
%F From _Ilya Gutkovskiy_, Jul 28 2016: (Start)
%F Inverse binomial transform of A084130.
%F Exponential convolution of A000079 and A084058.
%F Sum_{n>=0} (-1)^n*a(n)/n! = cosh(2*sqrt(2))/exp(3) = 0.4226407909842764637... (End)
%F a(2*n+1) = 2*a(n)*a(n+1) - 3. - _Timothy L. Tiffin_, Oct 12 2016
%F a(n) = a(-n) for all n in Z. - _Michael Somos_, Jan 20 2017
%F a(2^n) = A001601(n+1). - _A.H.M. Smeets_, May 28 2017
%F a(A298210(n)) = A002350(2*n^2). - _A.H.M. Smeets_, Jan 25 2018
%F a(n) = S(n, 6) - 3*S(n-1, 6), for n >= 0, with S(n, 6) = A001109(n+1), (Chebyshev S of A049310). See the first comment and the formula a(n) = T(n, 3). - _Wolfdieter Lang_, Nov 22 2020
%F From _Peter Bala_, Dec 31 2021: (Start)
%F a(n) = [x^n] (3*x + sqrt(1 + 8*x^2))^n.
%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) hold for all prime p and positive integers n and k.
%F O.g.f. A(x) = 1 + x*d/dx(log(B(x))), where B(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. of A001850. (End)
%F From _Peter Bala_, Aug 17 2022: (Start)
%F Sum_{n >= 1} 1/(a(n) - 2/a(n)) = 1/2.
%F Sum_{n >= 1} (-1)^(n+1)/(a(n) + 1/a(n)) = 1/4.
%F Sum_{n >= 1} 1/(a(n)^2 - 2) = 1/2 - 1/sqrt(8). (End)
%e 99^2 + 99^2 = 140^2 + 2. - _Carmine Suriano_, Jan 05 2015
%e G.f. = 1 + 3*x + 17*x^2 + 99*x^3 + 577*x^4 + 3363*x^5 + 19601*x^6 + 114243*x^7 + ...
%p a[0]:=1: a[1]:=3: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # _Zerinvary Lajos_, Jul 26 2006
%p A001541:=-(-1+3*z)/(1-6*z+z**2); # _Simon Plouffe_ in his 1992 dissertation
%t Table[Simplify[(1/2) (3 + 2 Sqrt[2])^n + (1/2) (3 - 2 Sqrt[2])^n], {n, 0, 20}] (* _Artur Jasinski_, Feb 10 2010 *)
%t a[ n_] := If[n == 0, 1, With[{m = Abs @ n}, m Sum[4^i Binomial[m + i, 2 i]/(m + i), {i, 0, m}]]]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := ChebyshevT[ n, 3]; (* _Michael Somos_, Jul 11 2011 *)
%t LinearRecurrence[{6, -1}, {1, 3}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)
%o (PARI) {a(n) = real((3 + quadgen(32))^n)}; /* _Michael Somos_, Apr 07 2003 */
%o (PARI) {a(n) = subst( poltchebi( abs(n)), x, 3)}; /* _Michael Somos_, Apr 07 2003 */
%o (PARI) {a(n) = if( n<0, a(-n), polsym(1 - 6*x + x^2, n) [n+1] / 2)}; /* _Michael Somos_, Apr 07 2003 */
%o (PARI) {a(n) = polchebyshev( n, 1, 3)}; /* _Michael Somos_, Jul 11 2011 */
%o (PARI) a(n)=([1,2,2;2,1,2;2,2,3]^n)[3,3] \\ _Vim Wenders_, Mar 28 2007
%o (Magma)[n: n in [1..10000000] |IsSquare(8*(n^2-1))] // _Vincenzo Librandi_, Nov 18 2010]
%o (Haskell)
%o a001541 n = a001541_list !! (n-1)
%o a001541_list =
%o 1 : 3 : zipWith (-) (map (* 6) $ tail a001541_list) a001541_list
%o -- _Reinhard Zumkeller_, Oct 06 2011
%o (Scheme, with memoization-macro definec)
%o (definec (A001541 n) (cond ((zero? n) 1) ((= 1 n) 3) (else (- (* 6 (A001541 (- n 1))) (A001541 (- n 2))))))
%o ;; _Antti Karttunen_, Oct 04 2016
%Y Bisection of A001333. A003499(n) = 2a(n).
%Y Cf. A055997 = numbers n such that n(n-1)/2 is a square.
%Y Row 1 of array A188645.
%Y Cf. A046090, A001109, A053142, A084130, A001601, A056771, A077420, A005319, A082532, A001542.
%Y Cf. A055792 (terms squared), A132592.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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