A077289 Triangular numbers that are 1/6 of another triangular number.

0, 1, 6, 105, 595, 10296, 58311, 1008910, 5713890, 98862891, 559902916, 9687554415, 54864771885, 949281469786, 5376187741821, 93019896484620, 526811533926580, 9115000574022981, 51622154137063026, 893177036357767525, 5058444293898249975, 87522234562487194476

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

The triangular numbers multiplied by 6 are in A077290.

Examples

			b(3)=14 so a(3) = 14*15/2 = 105, etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
• Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Crossrefs

Cf. A077288, A077290, A077291.

Programs

Maple

f := gfun:-rectoproc({a(-2) = 1, a(-1) = 0, a(0) = 0, a(1) = 1, a(n) = 98*a(n-2)-a(n-4)+7}, a(n), remember); map(f, [`$`(0 .. 1000)])[]; # Vladimir Pletser, Feb 19 2021

Mathematica

tr6Q[n_]:= IntegerQ[1/2 (Sqrt[1+48n]-1)]; Select[Accumulate[ Range[0,1380000]],tr6Q]  (* Harvey P. Dale, Apr 21 2011 *)

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\6, ", ") ) );
\\ Joerg Arndt, Jul 03 2013

PARI

concat(0, Vec(-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

Formula

Let b(n) be A077288. Then a(n)=b(n)*(b(n)+1)/2.

G.f.: -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-4) + 7. - Vladimir Pletser, Feb 19 2021

96*a(n) = 9*A072256(n+1) -2*(-1)^n*A054320(n) -7. - R. J. Mathar, Oct 01 2021