A383225 a(n) = sqrt(1 + P(n)*P(n+1)*P(n+2)*P(n+3)) where P(n) = A000129(n) are the Pell numbers.

1, 11, 59, 349, 2029, 11831, 68951, 401881, 2342329, 13652099, 79570259, 463769461, 2703046501, 15754509551, 91824010799, 535189555249, 3119313320689, 18180690368891, 105964828892651, 617608282987021, 3599684869029469, 20980500931189799, 122283320718109319, 712719423377466121

Offset

0,2

Comments

The ratios a(n+1)/a(n) converge to 2*sqrt(2)+3 (A156035).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Formula

a(n) = P(n+1)*P(n+2) - (-1)^n. [Corrected by Seiichi Manyama, May 25 2025]

G.f.: (1+6*x-x^2)/((1-6*x+x^2)*(1+x)). - Joerg Arndt, Apr 26 2025

Example

a(5) = sqrt(1 + 29*70*169*408) = 11831.

Mathematica

LinearRecurrence[{5, 5, -1}, {1, 11, 59}, 25] (* Amiram Eldar, Apr 26 2025 *)

Prog

(PARI) Vec((1+6*x-x^2)/((1-6*x+x^2)*(1+x))+O(x^25)) \\ Joerg Arndt, Apr 26 2025
(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = pell(n+1)*pell(n+2)-(-1)^n; \\ Seiichi Manyama, May 25 2025

Crossrefs

Cf. A000129, A156035.

Sequence in context: A249891 A186256 A164299 * A253207 A241860 A082884

Adjacent sequences: A383222 A383223 A383224 * A383226 A383227 A383228

Keyword

nonn,easy

Author

Jules Beauchamp, Apr 26 2025

Status

approved