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A076139 Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m) = T(k) for some k. 25
0, 1, 15, 210, 2926, 40755, 567645, 7906276, 110120220, 1533776805, 21362755051, 297544793910, 4144264359690, 57722156241751, 803965923024825, 11197800766105800, 155965244802456376, 2172315626468283465, 30256453525753512135, 421418033734080886426 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

LINKS

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).

FORMULA

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).

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)

a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3) with a(0)=0, a(1)=1, a(2)=15.

G.f.: x/(1-15*x+15*x^2-x^3). (End)

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).

G.f.: x / ((1 - x) * (1 - 14*x +x^2)).

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

EXAMPLE

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).

MATHEMATICA

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 *)

PROG

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

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

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

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

CROSSREFS

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).

Sequence in context: A112496 A000483 A162785 * A163091 A163440 A163962

Adjacent sequences: A076136 A076137 A076138 * A076140 A076141 A076142

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

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

Chebyshev comments from Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified December 9 17:12 EST 2022. Contains 358702 sequences. (Running on oeis4.)