A088210 - OEIS

%I #20 Feb 08 2024 09:46:14

%S 1,5,17,53,157,449,1253,3433,9273,24765,65529,172061,448853,1164409,

%T 3006157,7728337,19794545,50532469,128621281,326513669,826887693,

%U 2089505841,5269572021,13265211961,33336792745,83648953133,209591807177

%N 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], ...

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

%D 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.)

%H Paolo Xausa, <a href="/A088210/b088210.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,-1).

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

%F a(n) = A000129(n) + (n+1)*A000129(n+1) where A000129 are the Pell numbers. [Corrected by _Paolo Xausa_, Feb 08 2024]

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

%t LinearRecurrence[{4, -2, -4, -1}, {1, 5, 17, 53}, 30] (* _Paolo Xausa_, Feb 08 2024 *)

%Y Cf. A000129, A088211, A054459, A026937.

%K frac,nonn

%O 0,2

%A _Paul D. Hanna_, Sep 23 2003