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%I #103 Feb 05 2022 15:36:37
%S 1,2,4,9,21,50,120,289,697,1682,4060,9801,23661,57122,137904,332929,
%T 803761,1940450,4684660,11309769,27304197,65918162,159140520,
%U 384199201,927538921,2239277042,5406093004,13051463049,31509019101,76069501250,183648021600
%N a(n) = floor( a(n-1)/(sqrt(2) - 1) ), with a(0) = 1.
%C a(n) = A048739(n-1)+1 = 1/2 * (P(n)+P(n-1)+1), with P(n) = Pell numbers (A000129).
%C Number of (3412,#)-avoiding involutions in S_{n+1}, where # can be one of 22 patterns, see Egge reference.
%C Number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 1, s(n+1) = 1. - _Herbert Kociemba_, Jun 02 2004
%C Define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1} > a_{n+1}/a_n for n >= 0 . This is S(2,4). (For proof, see the Alekseyev link.) - _R. K. Guy_
%C This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211). - _Sameen Ahmed Khan_, Jun 28 2010
%C Partial sums of the Pell numbers prefaced with a 1: (1, 1, 2, 5, 12, 29, 70, ...). - _Gary W. Adamson_, Feb 15 2012
%C The number of ways to write an n-bit binary sequence and then give runs of ones weakly incrementing labels starting with 1, e.g., 0011010011022203003330044040055555. - _Andrew Woods_, Jan 03 2015
%C Sums of the positive coefficients in Chebyshev polynomials of the first kind, beginning with T_1. a(n+1)/a(n) approaches 1/(sqrt(2)-1). - _Gregory Gerard Wojnar_, Mar 19 2018
%H Clark Kimberling, <a href="/A024537/b024537.txt">Table of n, a(n) for n = 0..250</a>
%H Max Alekseyev, <a href="/A024537/a024537.txt">Notes on A024537</a>
%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2) (2000) 175-179. [From _Sameen Ahmed Khan_, Jun 28 2010]
%H Michael D. Barrus, <a href="https://arxiv.org/abs/1608.01358">Weakly threshold graphs</a>, arXiv preprint arXiv:1608.01358 [math.CO], 2016.
%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
%H E. S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations</a>, arXiv:math/0307050 [math.CO], 2003; section 8.
%H S. Felsner, D. Heldt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Felsner/felsner2.html">Lattice Path Enumeration and Toeplitz Matrices</a>, J. Int. Seq. 18 (2015) # 15.1.3.
%H Daniel Heldt, <a href="https://depositonce.tu-berlin.de/bitstream/11303/5553/5/heldt_daniel.pdf">On the mixing time of the face flip-and up/down Markov chain for some families of graphs</a>, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346 [physics.gen-ph], 2010. [_Sameen Ahmed Khan_, Jun 28 2010]
%H J. V. Leyendekkers and A. G. Shannon, <a href="http://nntdm.net/volume-18-2012/number-2/58-62/">Pellian sequence relationships among pi, e, sqrt(2)</a>, Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See {u_n}. - _N. J. A. Sloane_, Dec 23 2012
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F a(n) = 2*a(n-1) + a(n-2) - 1. - _Christian G. Bower_
%F a(n) = 3*a(n-1) - a(n-2) - a(n-3).
%F From _Paul Barry_, Dec 25 2003: (Start)
%F G.f.: (1 - x - x^2)/((1-x)*(1 - 2*x - x^2)) = (1 - x - x^2)/(1 - 3*x + x^2 + x^3).
%F E.g.f.: exp((1+sqrt(2))*x)*(1+sqrt(2))/4+exp((1-sqrt(2))*x)*(1-sqrt(2))/4+exp(x)/2. (End)
%F a(n) = (1/4)*(2 + (1-sqrt(2))^(n+1) + (1+sqrt(2))^(n+1)). - _Herbert Kociemba_, Jun 02 2004
%F Let M = a tridiagonal matrix with all 1's in the super and main diagonals and [1,1,0,0,0,...] in the subdiagonal, and let V = vector [1,0,0,0,...], and the rest zeros. The sequence is generated as the leftmost column from iterates of M*V. - _Gary W. Adamson_, Jun 07 2011
%F G.f.: (1 + Q(0)*x/2)/(1-x), where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 06 2013
%F a(n) = A171842(n+1), n>=0. That sequence starts with an extra 1. - _Andrew Woods_, Jan 03 2015
%F a(n) = 1 + sum_{k=1..floor((n+1)/2)} C(n+1,2*k)*2^(k-1). - _Andrew Woods_, Jan 03 2015
%t NestList[Floor[#/(Sqrt[2]-1)]&,1,40] (* _Harvey P. Dale_, Apr 01 2012 *)
%t LinearRecurrence[{3, -1, -1}, {1, 2, 4}, 31] (* _Jean-François Alcover_, Jan 07 2019 *)
%o (PARI) a=vector(99);a[1]=1; for(n=2,#a,a[n]=a[n-1]\(sqrt(2) - 1)); a \\ _Charles R Greathouse IV_, Jun 14 2011
%o (PARI) x='x+O('x^99); Vec((1-x-x^2)/((1-x)*(1-2*x-x^2))) \\ _Altug Alkan_, Mar 19 2018
%Y Cf. A048211, A153588, A174283-A174286, A176499-A176502. - _Sameen Ahmed Khan_, Jun 28 2010
%Y Cf. A171842. - _Andrew Woods_, Jan 03 2015
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
%E Edited by _N. J. A. Sloane_ at the suggestion of _Max Alekseyev_, Aug 24 2007
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