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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. 17
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

Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.

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).

a(0)=0, a(1)=1, a(2)=15; a(n) = 15*a(n-1)-15*a(n-2)+a(n-3) for n>=3. G.f.: x/(1-15*x+15*x^2-x^3). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

a(n+1) = sum(S(k, 14), k=0..n), n>=0, with 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

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-Fran├žois 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

CROSSREFS

The m values are in A061278, the k values are in A001571.

Cf. A014979, A076140, A108281.

Cf. A045899, A123480, A217855.

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 September 22 16:56 EDT 2019. Contains 327311 sequences. (Running on oeis4.)