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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], ...
3
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.)
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
KEYWORD
frac,nonn
AUTHOR
Paul D. Hanna, Sep 23 2003
STATUS
approved