A322708 a(0)=0, a(1)=6 and a(n) = 26*a(n-1) - a(n-2) + 12 for n > 1.

0, 6, 168, 4374, 113568, 2948406, 76545000, 1987221606, 51591216768, 1339384414374, 34772403556968, 902743108066806, 23436548406180000, 608447515452613206, 15796198853361763368, 410092722671953234374, 10646614590617422330368, 276401886633381027355206

Offset: 0

Author

Seiichi Manyama, Dec 24 2018

Keywords

• nonn
• easy

Comments

Solutions to X*(X+1)=42*Y^2 with Y=A097309. - R. J. Mathar, Mar 14 2023

Examples

			(sqrt(7) + sqrt(6))^2 = 13 + 2*sqrt(42) = sqrt(169) + sqrt(168). So a(2) = 168.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (27,-27,1).

Crossrefs

Row 6 of A322699.

Programs

Mathematica

LinearRecurrence[{27,-27,1},{0,6,168},20] (* Harvey P. Dale, Apr 30 2022 *)

PARI

concat(0, Vec(6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)) + O(x^20))) \\ Colin Barker, Dec 24 2018

Formula

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(7) + sqrt(6))^n.

sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(7) - sqrt(6))^n.

a(n) = 27*a(n-1) - 27*a(n-2) + a(n-3) for n > 2.

From Colin Barker, Dec 24 2018: (Start)

G.f.: 6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)).

a(n) = ((13+2*sqrt(42))^(-n) * (-1+(13+2*sqrt(42))^n)^2) / 4.

(End)

2*a(n) = A097308(n)-1. - R. J. Mathar, Mar 14 2023