A108281 -id:A108281 - cp's OEIS frontend

A076139 Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m) = T(k) for some k.

0, 1, 15, 210, 2926, 40755, 567645, 7906276, 110120220, 1533776805, 21362755051, 297544793910, 4144264359690, 57722156241751, 803965923024825, 11197800766105800, 155965244802456376, 2172315626468283465, 30256453525753512135, 421418033734080886426

Offset: 0

Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002

Both triangular and generalized pentagonal numbers: intersection of A000217 and A001318. - Vladeta Jovovic, Aug 29 2004

Partial sums of Chebyshev polynomials S(n,14).

			G.f. = x + 15*x^2 + 210*x^3 + 2926*x^4 + 40755*x^5 + 567645*x^6 + ...
a(3)=210=T(20) and 3*210=630=T(35).

Colin Barker, Table of n, a(n) for n = 0..874
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

The m values are in A061278, the k values are in A001571.
Cf. A014979, A076140, A108281.
Cf. A045899, A123480, A217855.
Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

[(Evaluate(ChebyshevU(n+1), 7) - Evaluate(ChebyshevU(n), 7) - 1)/12 : n in [0..30]]; // G. C. Greubel, Feb 03 2022

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 *)

{a(n) = polchebyshev( n, 2, 7) / 14 + polchebyshev( n, 1, 7)/ 84 - 1 / 12}; /* Michael Somos, Jun 16 2011 */

concat(0, Vec(-x/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015

[(chebyshev_U(n,7) - chebyshev_U(n-1,7) - 1)/12 for n in (0..30)] # G. C. Greubel, Feb 03 2022

G.f.: x / ((1 - x) * (1 - 14*x +x^2)).
a(n+1) = Sum_{k=0..n} S(k, 14), n >= 0, where S(k, 14) = U(k, 7) = A007655(k+2).
a(n+1) = (S(n+1, 14) - S(n, 14) - 1)/12, n >= 0.
a(n) = 14 * a(n-1) - a(n-2) + 1. a(0)=0, a(1)=1.
a(-n) = a(n-1).
a(n) = A061278(n)*(A061278(n)+1)/2.
a(n) = (1/288)*(-24 + (12-6*sqrt(3))*(7-4*sqrt(3))^n + (12+6*sqrt(3))*(7+4*sqrt(3))^n).
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
a(2*n) = A108281(n + 1). a(2*n + 1) = A014979(n + 2). - Michael Somos, Jun 16 2011
a(n) = (1/2)*A217855(n) = (1/3)*A076140(n) = (1/4)*A123480(n) = (1/8)*A045899(n). - Peter Bala, Dec 31 2012
a(n) = A001353(n) * A001353(n-1) / 4. - Richard R. Forberg, Aug 26 2013
a(n) = ((2+sqrt(3))^(2*n+1) + (2-sqrt(3))^(2*n+1))/48 - 1/12. - Vladimir Pletser, Jan 15 2021

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

Chebyshev comments from Wolfdieter Lang, Aug 31 2004

A014979 Numbers that are both triangular and pentagonal.

Original entry on oeis.org

0, 1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, 421418033734080886426, 81752926228785223683195, 15859646270350599313653420

Offset: 1

Glenn Johnston (glennj(AT)sonic.net)

			G.f. = x^2 + 210*x^3 + 40755*x^4 + 7906276*x^5 + 1533776805*x^6 + ...
a(4) = 40755 which is 285*(285-1)/2 = 165*(3*165-1)/2.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 210, p. 61, Ellipses, Paris 2008.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 22.

C. Gill, solution to question no. 8, Mathematical Miscellany, 1 (1836), pp. 220-225, at p. 223.
J. C. Su, On some properties of two simultaneous polygonal sequences, JIS 10 (2007) 07.10.4, example 4.2.
Eric Weisstein's World of Mathematics, Pentagonal Triangular Number.
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Cf. A046174, A046175, A076139, A108281.

a[ n_] := ChebyshevU[ 2 n - 3, 7] / 14 + ChebyshevT[ 2 n - 3, 7] / 84 - 1/12; (* Michael Somos, Feb 24 2015 *)
LinearRecurrence[{195,-195,1},{0,1,210},20] (* Harvey P. Dale, May 19 2017 *)

{a(n) = polchebyshev( 2*n - 3, 2, 7) / 14 + polchebyshev( 2*n - 3, 1, 7) / 84 - 1 / 12}; /* Michael Somos, Jun 16 2011 */

a(n) = 194 * a(n-1) - a(n-2) + 16.
G.f.: x^2 * (1 + 15*x) / ((1 - x) * (1 - 194*x + x^2)).
a(n)=((((1+sqrt(3))^(4*n-1)-(1-sqrt(3))^(4*n-1))/(2^(2*n+1)*sqrt(3)))^2)/2-1/8. - John Sillcox (johnsillcox(AT)hotmail.com), Sep 01 2003
a(n+1) = 97*a(n)+8+7*(192*a(n)^2+32*a(n)+1)^(1/2) - Richard Choulet, Sep 19 2007
a(n) = A076139(2*n - 3) = A108281(2 - n). for all n in Z. - Michael Somos, Jun 16 2011

Corrected and extended by Warut Roonguthai

Edited by N. J. A. Sloane, Jul 24 2006

A225839 Triangular numbers representable as triangular(m) + triangular(2m).

Original entry on oeis.org

0, 378, 17766, 39209940, 1842032556, 4065365016846, 190985619471570, 421505175637435176, 19801770996209306328, 43702499616375188919330, 2053087220237987679246270, 4531162564803507161896556028, 212868189148913267563402477956, 469799997000254729943383533193910

Offset: 1

Alex Ratushnyak, May 17 2013

Triangular numbers of the sequence A147875: a(n) = A147875(A225785(n)) - see also Ralf Stephan in Program lines. [Bruno Berselli, May 20 2013]

Index entries for linear recurrences with constant coefficients, signature (1,103682,-103682,-1,1).

Cf. A000217, A147875.
Cf. A108281 (triangular numbers representable as triangular(m) + m^2).
Cf. A225785 (numbers n such that triangular(n) + triangular(2n) is a triangular number).

CoefficientList[Series[378 x (1 + 46 x + x^2)/((1 - x) (1 - 322 x + x^2) (1 + 322 x + x^2)), {x, 0, 20}], x] (* Bruno Berselli, May 20 2013 *)
LinearRecurrence[{1,103682,-103682,-1,1},{0,378,17766,39209940,1842032556},20] (* Harvey P. Dale, Jan 16 2019 *)

for(n=1,10^9,t=n*(5*n+3)/2;x=sqrtint(2*t);if(t==x*(x+1)/2,print(t))) \\ Ralf Stephan, May 17 2013

G.f.: 378*x*(1+46*x+x^2)/((1-x)*(1-322*x+x^2)*(1+322*x+x^2)). [Bruno Berselli, May 20 2013]

More terms from Bruno Berselli, May 20 2013

cp's OEIS Frontend

A076139 Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m) = T(k) for some k.

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A014979 Numbers that are both triangular and pentagonal.

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A225839 Triangular numbers representable as triangular(m) + triangular(2m).

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