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%I #208 Jul 27 2023 19:21:41
%S 1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,
%T 0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,
%U 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0
%N a(n) = 1 if n is a triangular number, otherwise 0.
%C This is essentially the q-expansion of the Jacobi theta function theta_2(q). (In theta_2 one has to ignore the initial factor of 2*q^(1/4) and then replace q by q^(1/2). See also A005369.) - _N. J. A. Sloane_, Aug 03 2014
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Ramanujan's theta function f(a, b) = Sum_{n=-inf..inf} a^(n*(n+1)/2) * b^(n*(n-1)/2).
%C This sequence is the concatenation of the base-b digits in the sequence b^n, for any base b >= 2. - Davis Herring (herring(AT)lanl.gov), Nov 16 2004
%C Number of partitions of n into distinct parts such that the greatest part equals the number of all parts, see also A047993; a(n)=A117195(n,0) for n > 0; a(n) = 1-A117195(n,1) for n > 1. - _Reinhard Zumkeller_, Mar 03 2006
%C Triangle T(n,k), 0 <= k <= n, read by rows, given by A000007 DELTA A000004 where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 03 2009
%C Convolved with A000041 = A022567, the convolution square of A000009. - _Gary W. Adamson_, Jun 11 2009
%C A008441(n) = Sum_{k=0..n} a(k)*a(n-k). - _Reinhard Zumkeller_, Nov 03 2009
%C Polcoeff inverse with alternate signs = A006950: (1, 1, 1, 2, 3, 4, 5, 7, ...). - _Gary W. Adamson_, Mar 15 2010
%C This sequence is related to Ramanujan's two-variable theta functions because this sequence is also the characteristic function of generalized hexagonal numbers. - _Omar E. Pol_, Jun 08 2012
%C Number 3 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - _Michael Somos_, May 04 2016
%C Number of partitions of n into consecutive parts that contain 1 as a part, n >= 1. - _Omar E. Pol_, Nov 27 2020
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
%D M. D. Hirschhorn, The Power of q, Springer, 2017. See Psi page 9.
%D J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
%D E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.
%H Reinhard Zumkeller, <a href="/A010054/b010054.txt">Table of n, a(n) for n = 0..10000</a>
%H Mohammad K. Azarian, <a href="https://doi.org/10.12988/imf.2022.912321">Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions</a>, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Conjecture 4.4, p. 137.
%H S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares</a>, Discrete Math. 274 (2004), no. 1-3, 9-24. See psi(q).
%H Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%H M. D. Hirschhorn and J. A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Sellers/sellers32.html">A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four</a>, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94, Proposition 1.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Michael Somos, <a href="/A108483/a108483.pdf">A Multisection of q-Series</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Wolfram Challenges, <a href="https://challenges.wolframcloud.com/challenge/separate-ones-by-zeros">Separate Ones by Zeroes</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F Expansion of f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%F Expansion of q^(-1) * (phi(q) - phi(q^4)) / 2 in powers of q^8. - _Michael Somos_, Jul 01 2014
%F Expansion of q^(-1/8) * eta(q^2)^2 / eta(q) in powers of q. - _Michael Somos_, Apr 13 2005
%F Euler transform of period 2 sequence [ 1, -1, ...]. - _Michael Somos_, Mar 24 2003
%F Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u6^3 + u2*u3^3 - u1*u2^2*u6. - _Michael Somos_, Apr 13 2005
%F a(n) = b(8*n + 1) where b()=A098108() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p > 2. - _Michael Somos_, Jun 06 2005
%F a(n) = A005369(2*n). - _Michael Somos_, Apr 29 2003
%F G.f.: theta_2(sqrt(q)) / (2 * q^(1/8)).
%F G.f.: 1 / (1 - x / (1 + x / (1 + x^1 / (1 - x / (1 + x / (1 + x^2 / (1 - x / (1 + x / (1 + x^3 / ...))))))))). - _Michael Somos_, May 11 2012
%F G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - _Vladeta Jovovic_, May 02 2002
%F a(0)=1; for n>0, a(n) = A002024(n+1)-A002024(n). - _Benoit Cloitre_, Jan 05 2004
%F G.f.: Sum_{j>=0} Product_{k=0..j} x^j. - _Jon Perry_, Mar 30 2004
%F a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2). - _Carl R. White_, Mar 18 2006
%F a(n) = round(sqrt(2n+1)) - round(sqrt(2n)). - _Hieronymus Fischer_, Aug 06 2007
%F a(n) = ceiling(2*sqrt(2n+1)) - floor(2*sqrt(2n)) - 1. - _Hieronymus Fischer_, Aug 06 2007
%F a(n) = f(n,0) with f(x,y) = if x > 0 then f(x-y,y+1), otherwise 0^(-x). - _Reinhard Zumkeller_, Sep 27 2008
%F a(n) = A035214(n) - 1.
%F From _Mikael Aaltonen_, Jan 22 2015: (Start)
%F Since the characteristic function of s-gonal numbers is given by floor(sqrt(2n/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)) - floor(sqrt(2(n-1)/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)), by setting s = 3 we get the following: For n > 0, a(n) = floor(sqrt(2*n+1/4)-1/2) - floor(sqrt(2*(n-1)+1/4)-1/2).
%F (End)
%F a(n) = (-1)^n * A106459(n). - _Michael Somos_, May 04 2016
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(-1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A002448. - _Michael Somos_, May 05 2016
%F G.f.: Sum_{n >= 0} x^(n*(n+1)/2) = Product_{n >= 1} (1 - x^n)*(1 + x^n)^2 = Product_{n >= 1} (1 - x^(2*n))*(1 + x^n) = Product_{n >= 1} (1 - x^(2*n))/(1 - x^(2*n-1)). From the sum and product representations of theta_2(0, sqrt(q))/(2*q^(1/8))) function. The last product, given by _Vladeta Jovovic_ above, is obtained from the second to last one by an Euler identity, proved via f(x) := Product_{n >= 1} (1 - x^(2*n-1))*Product_{n >= 1} (1 + x^n) = f(x^2), by moving the odd-indexed factors of the second product to the first product. This leads to f(x) = f(0) = 1. - _Wolfdieter Lang_, Jul 05 2016
%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 08 2017
%e G.f. = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 + x^45 + x^55 + x^66 + ...
%e G.f. for B(q) = q * A(q^8): q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
%e From _Philippe Deléham_, Jan 04 2008: (Start)
%e As a triangle this begins:
%e 1;
%e 1, 0;
%e 1, 0, 0;
%e 1, 0, 0, 0;
%e 1, 0, 0, 0, 0;
%e 1, 0, 0, 0, 0, 0;
%e ... (End)
%p A010054 := proc(n)
%p if issqr(1+8*n) then
%p 1;
%p else
%p 0;
%p end if;
%p end proc:
%p seq(A010054(n),n=0..80) ; # _R. J. Mathar_, Feb 22 2021
%t a[ n_] := SquaresR[ 1, 8 n + 1] / 2; (* _Michael Somos_, Nov 15 2011 *)
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^2], {x, 0, n + Floor @ Sqrt[n]}] // Normal // TrigToExp) /. {y -> x}, {x, 0, n}]]; (* _Michael Somos_, Nov 15 2011 *)
%t Table[If[IntegerQ[(Sqrt[8n+1]-1)/2],1,0],{n,0,110}] (* _Harvey P. Dale_, Oct 29 2012 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]; (* _Michael Somos_, Jul 01 2014 *)
%t Module[{tr=Accumulate[Range[20]]},If[MemberQ[tr,#],1,0]&/@Range[Max[tr]]] (* _Harvey P. Dale_, Mar 13 2023 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A), n))}; /* _Michael Somos_, Mar 14 2011 */
%o (PARI) {a(n) = issquare( 8*n + 1)}; /* _Michael Somos_, Apr 27 2000 */
%o (PARI) a(n) = ispolygonal(n, 3); \\ _Michel Marcus_, Jan 22 2015
%o (Haskell)
%o a010054 = a010052 . (+ 1) . (* 8)
%o a010054_list = concatMap (\x -> 1 : replicate x 0) [0..]
%o -- _Reinhard Zumkeller_, Feb 12 2012, Oct 22 2011, Apr 02 2011
%o (Magma) Basis( ModularForms( Gamma0(16), 1/2), 362) [2] ; /* _Michael Somos_, Jun 10 2014 */
%o (Python)
%o from sympy import integer_nthroot
%o def A010054(n): return int(integer_nthroot((n<<3)+1,2)[1]) # _Chai Wah Wu_, Nov 15 2022
%o (Sage) # uses[EulerTransform from A166861]
%o b = BinaryRecurrenceSequence(-1, 0)
%o a = EulerTransform(b)
%o print([a(n) for n in range(88)]) # _Peter Luschny_, Nov 17 2022
%o (Clojure)
%o (def A010054 (mapcat #(cons 1 (replicate % 0)) (range))) ; _Tony Zorman_, Apr 03 2023
%Y Cf. A000217, A002448, A005369, A023531, A035214, A022567, A052343, A006950, A106459, A127648.
%Y Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
%Y Cf. A106507 (reciprocal series).
%K nonn,tabl,easy
%O 0,1
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, Apr 27 2000
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