A088210 Numerators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,...(n 2's)...,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], ...

1, 5, 17, 53, 157, 449, 1253, 3433, 9273, 24765, 65529, 172061, 448853, 1164409, 3006157, 7728337, 19794545, 50532469, 128621281, 326513669, 826887693, 2089505841, 5269572021, 13265211961, 33336792745, 83648953133, 209591807177

Offset: 0

Author

Paul D. Hanna, Sep 23 2003

Keywords

• frac
• nonn

Comments

Denominators are A088211. Partial sums form A054459. Second differences form A026937.

Examples

			a(3)/A088211(3) = [2;2,2,4] = 53/22.
		

References

• R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (See the foot of page 136.)

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Crossrefs

Cf. A000129, A088211, A054459, A026937.

Programs

Mathematica

LinearRecurrence[{4, -2, -4, -1}, {1, 5, 17, 53}, 30] (* Paolo Xausa, Feb 08 2024 *)

Formula

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

a(n) = A000129(n) + (n+1)*A000129(n+1) where A000129 are the Pell numbers. [Corrected by Paolo Xausa, Feb 08 2024]