A200998 Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4*T(m)=3*T(k) for some k.

0, 21, 4095, 794430, 154115346, 29897582715, 5799976931385, 1125165627105996, 218276331681631860, 42344483180609474865, 8214611460706556491971, 1593592278893891349967530, 309148687493954215337208870, 59973251781548223884068553271

Offset: 0

Author

Charlie Marion, Dec 20 2011

Keywords

• nonn
• easy

Comments

For n>1, a(n) = 194*a(n-1) - a (n-2) + 21. See A200993 for generalization.

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 0..425
• Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Crossrefs

Cf. A001652, A029549, A053141, A075528, A200993-A201008.

Programs

Magma

I:=[0,21]; [n le 2 select I[n] else  194*Self(n-1) - Self(n-2) + 21: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

Mathematica

LinearRecurrence[{195, -195, 1}, {0, 21, 4095}, 30] (* Vincenzo Librandi, Mar 03 2016 *)

PARI

concat(0, Vec(21/(1 - 195*x + 195*x^2 - x^3) + O(x^99))) \\ Charles R Greathouse IV, Dec 20 2011

Formula

G.f.: (21*x)/(1 - 195*x + 195*x^2 - x^3).

From Colin Barker, Mar 02 2016: (Start)

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

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))/128.

(End)