A200999 Triangular numbers, T(m), that are four-thirds of another triangular number; T(m) such that 3*T(m) = 4*T(k) for some k.

0, 28, 5460, 1059240, 205487128, 39863443620, 7733302575180, 1500220836141328, 291035108908842480, 56459310907479299820, 10952815280942075322628, 2124789705191855133290040, 412198249991938953782945160, 79964335708730965178758071028

Offset: 0

Author

Charlie Marion, Feb 15 2012

Keywords

• nonn
• easy

Comments

Numbers h such that 6*h+1 and 8*h+1 are both squares. [Bruno Berselli, Jul 07 2014]

Examples

			3*0 = 4*0.
3*28 = 4*21.
3*5640 = 4*4095.
3*1059240 = 4*794430.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Crossrefs

Cf. A200994, A245031.

Programs

Mathematica

LinearRecurrence[{195, -195, 1}, {0, 28, 5460}, 20] (* T. D. Noe, Feb 15 2012 *)

PARI

concat(0, Vec(28*x/((1-x)*(1-194*x+x^2)) + O(x^15))) \\ Colin Barker, Mar 02 2016

Formula

For n>1, a(n) = 194*a(n-1) - a (n-2) + 28. See A200998 for generalization.

From Colin Barker, Mar 02 2016: (Start)

a(n) = ((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(-7+4*sqrt(3)+(7+4*sqrt(3))*(97+56*sqrt(3))^n))/96.

a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>2.

G.f.: 28*x / ((1-x)*(1-194*x+x^2)).

(End)