A077400 Triangular numbers that are 7 times triangular numbers.

0, 21, 105, 5460, 26796, 1386945, 6806205, 352278696, 1728749400, 89477401965, 439095541521, 22726907820540, 111528538797060, 5772545109015321, 28327809758911845, 1466203730782071120, 7195152150224811696, 372409975073537049285, 1827540318347343259065

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..831

Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 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 linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Formula

Let b(n) = A077399(n) then a(n) = 7*b(n).

G.f.: -21*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)). - Colin Barker, Jul 02 2013

From _Vladimir Pletser, Feb 21 2021: (Start)

a(n) = 254*a(n - 2) - a (n - 4) + 126.

a(n) = a(n - 1) + 254*(a(n - 2) - a(n - 3)) - (a (n - 4) - a(n - 5)). (End)

Maple

f := gfun:-rectoproc({a(-2) = 21, a(-1) = 0, a(0) = 0, a(1) = 21, a(n) = 254*a(n-2)-a(n-4)+126}, a(n), remember): map(f, [`$`(0 .. 1000)])[] # Vladimir Pletser, Feb 21 2021

Mathematica

LinearRecurrence[{1, 254, -254, -1, 1}, {0, 21, 105, 5460, 26796}, 20] (* Harvey P. Dale, Oct 28 2013 *)

Prog

(PARI) concat(0, Vec(-21*x*(x^2+4*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 30); [0] cat Coefficients(R!(-21*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)))); // G. C. Greubel, Jan 18 2018

Crossrefs

Cf. A077398, A077399, A077401.

Sequence in context: A068986 A219631 A226742 * A041854 A219820 A071333

Adjacent sequences: A077397 A077398 A077399 * A077401 A077402 A077403

Keyword

easy,nonn

Author

Bruce Corrigan (scentman(AT)myfamily.com), Nov 05 2002

Status

approved