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%I A076139 #101 Feb 11 2026 16:47:38
%S A076139 0,1,15,210,2926,40755,567645,7906276,110120220,1533776805,
%T A076139 21362755051,297544793910,4144264359690,57722156241751,
%U A076139 803965923024825,11197800766105800,155965244802456376,2172315626468283465,30256453525753512135,421418033734080886426
%N A076139 Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m) = T(k) for some k.
%C A076139 Both triangular and generalized pentagonal numbers: intersection of A000217 and A001318. - _Vladeta Jovovic_, Aug 29 2004
%C A076139 Partial sums of Chebyshev polynomials S(n,14).
%H A076139 Colin Barker, <a href="/A076139/b076139.txt">Table of n, a(n) for n = 0..874</a>
%H A076139 Francesca Arici and Jens Kaad, <a href="https://arxiv.org/abs/2012.11186">Gysin sequences and SU(2)-symmetries of C*-algebras</a>, arXiv:2012.11186 [math.OA], 2020.
%H A076139 Roger B. Nelson, <a href="https://www.jstor.org/stable/10.4169/math.mag.89.3.159">Multi-Polygonal Numbers</a>, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
%H A076139 Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2020.
%H A076139 Vladimir Pletser, <a href="https://arxiv.org/abs/2102.12392">Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2102.12392 [math.GM], 2021.
%H A076139 Vladimir Pletser, <a href="https://arxiv.org/abs/2102.13494">Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations</a>, arXiv:2102.13494 [math.NT], 2021.
%H A076139 Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2021.
%H A076139 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%H A076139 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A076139 G.f.: x / ((1 - x) * (1 - 14*x +x^2)).
%F A076139 a(n+1) = Sum_{k=0..n} S(k, 14), n >= 0, where S(k, 14) = U(k, 7) = A007655(k+2).
%F A076139 a(n+1) = (S(n+1, 14) - S(n, 14) - 1)/12, n >= 0.
%F A076139 a(n) = 14 * a(n-1) - a(n-2) + 1. a(0)=0, a(1)=1.
%F A076139 a(-n) = a(n-1).
%F A076139 a(n) = A061278(n)*(A061278(n)+1)/2.
%F A076139 a(n) = (1/288)*(-24 + (12-6*sqrt(3))*(7-4*sqrt(3))^n + (12+6*sqrt(3))*(7+4*sqrt(3))^n).
%F A076139 a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3) with a(0)=0, a(1)=1, a(2)=15. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
%F A076139 a(2*n) = A108281(n + 1). a(2*n + 1) = A014979(n + 2). - _Michael Somos_, Jun 16 2011
%F A076139 a(n) = (1/2)*A217855(n) = (1/3)*A076140(n) = (1/4)*A123480(n) = (1/8)*A045899(n). - _Peter Bala_, Dec 31 2012
%F A076139 a(n) = A001353(n) * A001353(n-1) / 4. - _Richard R. Forberg_, Aug 26 2013
%F A076139 a(n) = ((2+sqrt(3))^(2*n+1) + (2-sqrt(3))^(2*n+1))/48 - 1/12. - _Vladimir Pletser_, Jan 15 2021
%F A076139 Sum_{n>=1} 1/a(n) = 8 - 4*sqrt(3) = 1 + A354128. - _Amiram Eldar_, Jan 26 2026
%F A076139 a(n) + a(n-1) = A098301(n) = A001353(n)^2. - _Klaus Purath_, Feb 01 2026
%e A076139 G.f. = x + 15*x^2 + 210*x^3 + 2926*x^4 + 40755*x^5 + 567645*x^6 + ...
%e A076139 a(3) = 210 = T(20) and 3*210 = 630 = T(35).
%t A076139 a[n_] := a[n] = 14*a[n-1] - a[n-2] + 1; a[0] = 0; a[1] = 1; Table[ a[n], {n, 0, 17}] (* _Jean-François Alcover_, Dec 15 2011, after given formula *)
%o A076139 (PARI) {a(n) = polchebyshev( n, 2, 7) / 14 + polchebyshev( n, 1, 7)/ 84 - 1 / 12}; /* _Michael Somos_, Jun 16 2011 */
%o A076139 (PARI) concat(0, Vec(-x/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ _Colin Barker_, May 15 2015
%o A076139 (SageMath) [(chebyshev_U(n,7) - chebyshev_U(n-1,7) - 1)/12 for n in (0..30)] # _G. C. Greubel_, Feb 03 2022
%o A076139 (Magma) [(Evaluate(ChebyshevU(n+1), 7) - Evaluate(ChebyshevU(n), 7) - 1)/12 : n in [0..30]]; // _G. C. Greubel_, Feb 03 2022
%Y A076139 The m values are in A061278, the k values are in A001571.
%Y A076139 Cf. A000217, A001318, A001353, A007655, A014979, A076140, A108281.
%Y A076139 Cf. A045899, A098301, A123480, A217855, A354128.
%Y A076139 Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
%K A076139 easy,nonn
%O A076139 0,3
%A A076139 Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002
%E A076139 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
%E A076139 Chebyshev comments from _Wolfdieter Lang_, Aug 31 2004

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