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

0,2

Comments

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

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).

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]

Example

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

Mathematica

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

Crossrefs

Cf. A000129, A088211, A054459, A026937.

Sequence in context: A158896 A294781 A110318 * A135344 A222160 A027028

Adjacent sequences: A088207 A088208 A088209 * A088211 A088212 A088213

Keyword

frac,nonn

Author

Paul D. Hanna, Sep 23 2003

Status

approved