A188930 -id:A188930 - cp's OEIS frontend

A322707 a(0)=0, a(1)=5 and a(n) = 22*a(n-1) - a(n-2) + 10 for n > 1.

0, 5, 120, 2645, 58080, 1275125, 27994680, 614607845, 13493377920, 296239706405, 6503780163000, 142786923879605, 3134808545188320, 68823001070263445, 1510971215000607480, 33172543728943101125, 728284990821747617280, 15989097254349504479045

Offset: 0

Seiichi Manyama, Dec 24 2018

Solutions to X*(X+1)=30*Y^2 with Y=A077421. - R. J. Mathar, Mar 14 2023

			(sqrt(6) + sqrt(5))^2 = 11 + 2*sqrt(30) = sqrt(121) + sqrt(120). So a(2) = 120.

Colin Barker, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

Row 5 of A322699.

Cf. A188930 (sqrt(5)+sqrt(6)).

concat(0, Vec(5*x*(1 + x) / ((1 - x)*(1 - 22*x + x^2)) + O(x^20))) \\ Colin Barker, Dec 24 2018

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(6) + sqrt(5))^n.
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(6) - sqrt(5))^n.
a(n) = 23*a(n-1) - 23*a(n-2) + a(n-3) for n > 2.
From Colin Barker, Dec 24 2018: (Start)
G.f.: 5*x*(1 + x) / ((1 - x)*(1 - 22*x + x^2)).
a(n) = ((11+2*sqrt(30))^(-n) * (-1+(11+2*sqrt(30))^n)^2) / 4.
(End)
2*a(n) = A077422(n)-1. - R. J. Mathar, Mar 16 2023

A188931 Continued fraction of sqrt(5)+sqrt(6).

Original entry on oeis.org

4, 1, 2, 5, 1, 1, 4, 1, 2, 24, 1, 2, 4, 2, 1, 3, 8, 3, 1, 2, 1, 1, 1, 40, 3, 1, 3, 1, 3, 3, 5, 14, 1, 5, 3, 6, 1, 9, 3, 1, 1, 10, 1, 2, 2, 1, 1, 23, 6, 4, 1, 1, 4, 1, 3, 21, 1, 3, 4, 1, 2, 1, 1, 1, 1, 4, 54, 8, 2, 3, 1, 6, 1, 6, 3, 2, 32, 1, 1, 4, 2, 1, 1, 2, 1, 1, 6, 1, 4, 2, 2, 1, 1, 1, 3, 14, 1, 11, 2, 3, 6, 32, 1, 1, 1, 8, 1, 1, 1, 3, 1, 2, 1, 10, 1, 4, 4, 1, 2, 1

Offset: 0

Clark Kimberling, Apr 13 2011

For a geometric interpretation, see A188640 and A188930.

			sqrt(5)+sqrt(6)=[4,1,2,5,1,1,4,1,2,24,...].

Cf. A188930 (decimal expansion).

r = 20^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[Sqrt[5]+Sqrt[6],120] (* Harvey P. Dale, Jun 26 2021 *)

Offset changed by Andrew Howroyd, Jul 08 2024

cp's OEIS Frontend

A322707 a(0)=0, a(1)=5 and a(n) = 22*a(n-1) - a(n-2) + 10 for n > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula