A322709 a(0)=0, a(1)=7 and a(n) = 30*a(n-1) - a(n-2) + 14 for n > 1.

0, 7, 224, 6727, 201600, 6041287, 181037024, 5425069447, 162571046400, 4871706322567, 145988618630624, 4374786852596167, 131097616959254400, 3928553721925035847, 117725514040791821024, 3527836867501829594887, 105717380511014096025600, 3167993578462921051173127

Offset: 0

Author

Seiichi Manyama, Dec 24 2018

Keywords

• nonn
• easy

Comments

Also numbers k such that 7*A000217(k) is a square. - Metin Sariyar, Nov 16 2019

Examples

			(sqrt(8) + sqrt(7))^2 = 15 + 2*sqrt(56) = sqrt(225) + sqrt(224). So a(2) = 224.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..650
• Index entries for linear recurrences with constant coefficients, signature (31,-31,1).

Crossrefs

Row 7 of A322699.

Cf. A188932 (sqrt(7)+sqrt(8)).

Programs

Magma

a:=[0,7]; [n le 2 select a[n] else 30*Self(n-1)-Self(n-2)+14: n in [1..18]]; // Marius A. Burtea, Nov 16 2019

Magma

R:=PowerSeriesRing(Integers(), 18); [0] cat Coefficients(R!(7*x*(1 + x) / ((1 - x)*(1-30*x + x^2))));  // Marius A. Burtea, Nov 16 2019

Mathematica

LinearRecurrence[{31,-31,1}, {0, 7, 224}, 18] (* Metin Sariyar, Nov 23 2019 *)

PARI

concat(0, Vec(7*x*(1 + x) / ((1 - x)*(1 - 30*x + x^2)) + O(x^20))) \\ Colin Barker, Dec 25 2018

Formula

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(8) + sqrt(7))^n.

sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(8) - sqrt(7))^n.

a(n) = 31*a(n-1) - 31*a(n-2) + a(n-3) for n > 2.

From Colin Barker, Dec 25 2018: (Start)

G.f.: 7*x*(1 + x) / ((1 - x)*(1 - 30*x + x^2)).

a(n) = ((15+4*sqrt(14))^(-n) * (-1+(15+4*sqrt(14))^n)^2) / 4.

(End)

E.g.f.: (1/4)*(-2*exp(x) + exp((15-4*sqrt(14))*x) + exp((15+4*sqrt(14))*x)). - Stefano Spezia, Nov 16 2019

2*a(n) = A068203(n)-1. - R. J. Mathar, Mar 16 2023