A188932 -id:A188932 - cp's OEIS frontend

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

Seiichi Manyama, Dec 24 2018

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

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

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

Row 7 of A322699.

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

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

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

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

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

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

A188933 Continued fraction of sqrt(7)+sqrt(8).

Original entry on oeis.org

5, 2, 9, 5, 2, 687, 6, 4, 1, 2, 2, 1, 2, 3, 1, 1, 1, 1, 1, 317, 19, 1, 1, 1, 6, 3, 8, 11, 10, 7, 1, 1, 5, 2, 1, 14, 1, 35, 6, 6, 1, 7, 2, 4, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 2, 84, 1, 3, 1, 6, 2, 1, 1, 2, 3, 1, 1, 7, 1, 4, 1, 31, 9, 1, 3, 9, 7, 1, 3, 1, 1, 3, 1, 1, 62, 12, 1, 24, 2, 1, 4, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 1, 3, 1, 10, 1, 1, 1, 14, 1, 1, 1, 2, 4, 1

Offset: 0

Clark Kimberling, Apr 13 2011

For a geometric interpretation, see A188640 and A188932.

			sqrt(7)+sqrt(8)=[5,2,9,5,2,687,6,4,1,2,2,1,2,3,1,1,1,1,1,...].

Cf. A188932 (decimal expansion).

r = 28^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Offset changed by Andrew Howroyd, Jul 08 2024

cp's OEIS Frontend

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

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