A097495 -id:A097495 - cp's OEIS frontend

A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

1, 1, 4, 8, 25, 65, 197, 571, 1753, 5351, 16746, 52626, 167547, 536559, 1732272, 5622960, 18357211, 60205319, 198323708, 655787680, 2176141555, 7244106347, 24185285341, 80960692691, 271685400443, 913784117809, 3079889039230

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform is the (1,8) Somos-4 sequence A097495(n+2).

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A001006.

Table[Sum[Binomial[n - k, k] 2^k * Hypergeometric2F1[(1 - #)/2, -#/2, 2, 4] &[n - 2 k], {k, 0, Floor[n/2]}], {n, 0, 26}] (* Michael De Vlieger, Feb 02 2017, after Peter Luschny at A001006 *)

G.f.: (1-x-2*x^2-sqrt(1-2*x-7*x^2+4*x^3+4*x^4))/(2*x^2) = (1/(1-2*x))*M(x/(1-2*x^2)), M(x) the g.f. of A010006.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * 2^k * A001006(n-2k).
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +7*(1-n)*a(n-2) +2*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012
a(0) = a(1) = 1; a(n) = a(n-1) + 2 * a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2). - Ilya Gutkovskiy, Nov 09 2021
a(n) ~ 17^(1/4) * (3 + sqrt(17))^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Nov 11 2021

A376024 a(0..4) = 1 and a(n) = (a(n-2)^2 + a(n-3)^2 + a(n-2)(3a(n-3) + a(n-4)) + a(n-1)*(a(n-3) - a(n-5)))/(a(n-4) + a(n-5)) for n > 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 11, 35, 83, 545, 2513, 13905, 152721, 1087873, 14651923, 238834051, 3135275371, 91466933731, 2155382231811, 63058059937761, 3261572372004353, 120654520736448833, 8395343248160222081, 661217270644238022305, 46110296193095128622723, 6786635441262507324649635

Offset: 0

Thomas Scheuerle, Sep 06 2024

nonn

An example of how a Somos recurrence can be generalized such that proving its integrality looks more difficult in the first glance. In this example the Somos-4 recurrence b(n) = (b(n-1) * b(n-3) + b(n-2)^2) / b(n-4) was modified by substitution of b(n-k) with (a(n-k) + a(n-k-1)).

This sequence is not a divisibility sequence unlike Somos-4 sequences are.

Cf. A006720, A097495 ( first 6 values coincidence with odd terms ).

a=vector(26); a[1]=a[2]=a[3]=a[4]=a[5]=1; for(n=6, #a, a[n]=(a[n-2]^2+a[n-3]^2+a[n-2]*(3*a[n-3]+a[n-4])+a[n-1]*(a[n-3]-a[n-5]))/(a[n-4]+a[n-5])); a

(a(n) + a(n+1))/2 = A006720(n).

cp's OEIS Frontend

A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

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A376024 a(0..4) = 1 and a(n) = (a(n-2)^2 + a(n-3)^2 + a(n-2)(3a(n-3) + a(n-4)) + a(n-1)*(a(n-3) - a(n-5)))/(a(n-4) + a(n-5)) for n > 4.

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cp's OEIS Frontend

A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A376024 a(0..4) = 1 and a(n) = (a(n-2)^2 + a(n-3)^2 + a(n-2)*(3*a(n-3) + a(n-4)) + a(n-1)*(a(n-3) - a(n-5)))/(a(n-4) + a(n-5)) for n > 4.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A376024 a(0..4) = 1 and a(n) = (a(n-2)^2 + a(n-3)^2 + a(n-2)(3a(n-3) + a(n-4)) + a(n-1)*(a(n-3) - a(n-5)))/(a(n-4) + a(n-5)) for n > 4.