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

Author

Jules Beauchamp, Apr 26 2025

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Crossrefs

Cf. A000129, A156035.

Programs

Mathematica

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

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

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