A077290 Triangular numbers that are 6 times other triangular numbers.

0, 6, 36, 630, 3570, 61776, 349866, 6053460, 34283340, 593177346, 3359417496, 58125326490, 329188631310, 5695688818716, 32257126450926, 558119378907720, 3160869203559480, 54690003444137886, 309732924822378156, 5359062218146605150, 30350665763389499850

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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 (1,98,-98,-1,1).

Formula

a(n) = 6*A077289(n).

G.f.: -6*x*(x^2+5*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)). - Colin Barker, Jul 02 2013

a(n) = 98*a(n-2) - a(n-1) + 42. - Vladimir Pletser, Feb 20 2021

Example

The k-th triangular number is T(k) = k*(k+1)/2, so T(35)/T(14) = (35*36/2)/(14*15/2) = 630/105 = 6, so T(35)=630 is a term. - Jon E. Schoenfield, Feb 20 2021

Maple

f := gfun:-rectoproc({a(-2) = 6, a(-1) = 0, a(0) = 0, a(1) = 6, a(n) = 98*a(n-2)-a(n-4)+42}, a(n), remember); map(f, [`$`(0 .. 1000)])[]; # Vladimir Pletser, Feb 20 2021

Mathematica

CoefficientList[Series[-6 x (x^2 + 5 x + 1)/((x - 1) (x^2 - 10 x + 1) (x^2 + 10 x + 1)), {x, 0, 20}], x] (* Michael De Vlieger, Apr 21 2021 *)

Prog

(PARI)
T(n)=n*(n+1)\2;
istriang(n)=issquare(8*n+1);
for(n=0, 10^10, t=T(n); if ( t%6==0 && istriang(t\6), print1(t, ", ") ) );
\\ Joerg Arndt, Jul 03 2013
(PARI) concat(0, Vec(-6*x*(x^2+5*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015

Crossrefs

Cf. A077288, A077289, A077291.

Subsequence of A000217.

Sequence in context: A367488 A339300 A296389 * A222780 A206636 A244841

Adjacent sequences: A077287 A077288 A077289 * A077291 A077292 A077293

Keyword

easy,nonn

Author

Bruce Corrigan (scentman(AT)myfamily.com), Nov 03 2002

Extensions

More terms from Joerg Arndt, Jul 03 2013

Status

approved