A077399 Triangular numbers that are 1/7 of triangular numbers.

0, 3, 15, 780, 3828, 198135, 972315, 50325528, 246964200, 12782485995, 62727934503, 3246701117220, 15932648399580, 824649301287903, 4046829965558835, 209457675826010160, 1027878878603544528, 53201425010505292755, 261077188335334751295, 13512952494992518349628

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..832

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) be A077398(n) then a(n) = b(n)*(b(n)+1)/2.

G.f.: -3*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)). - Colin Barker, Jul 02 2013

Comment from _Vladimir Pletser, Feb 21 2021: (Start)

a(n) = 254 a(n - 2) - a (n - 4) + 18.

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) = 3, a(-1) = 0, a(0) = 0, a(1) = 3, a(n) = 254*a(n-2)-a(n-4)+18}, a(n), remember); map(f, [`$`(0 .. 1000)])[] #Vladimir Pletser, Feb 21 2021

Mathematica

Select[Accumulate[Range[0, 5700000]], IntegerQ[(Sqrt[56#+1]-1)/2]&] (* Harvey P. Dale, Jan 18 2013 *)
LinearRecurrence[{1, 254, -254, -1, 1}, {0, 3, 15, 780, 3828}, 30] (* G. C. Greubel, Jan 18 2018 *)

Prog

(PARI) concat(0, Vec(-3*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!(-3*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, A077400, A077401.

Sequence in context: A217449 A167220 A296939 * A219122 A293541 A207332

Adjacent sequences: A077396 A077397 A077398 * A077400 A077401 A077402

Keyword

easy,nonn

Author

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

Status

approved