A033193 Binomial transform of A033192.

1, 2, 6, 19, 62, 207, 704, 2430, 8486, 29903, 106098, 378391, 1354700, 4863834, 17499302, 63055947, 227465414, 821215295, 2966571096, 10721076118, 38757594758, 140143505031, 506827217210, 1833150646599, 6630915738212, 23986989146162, 86775559512774, 313930265564035

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3. - Herbert Kociemba, Jun 16 2004

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1791
• N. J. A. Sloane, Transforms.
• Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Crossrefs

Cf. A033192.

Programs

Mathematica

CoefficientList[Series[(x^4 - 7 x^3 + 11 x^2 - 6 x + 1)/((1 - 3 x + x^2) (1 - 5 x + 5 x^2)), {x, 0, 23}], x] (* Michael De Vlieger, Feb 12 2022 *)

PARI

Vec((x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)) + O(x^24)) \\ Stefano Spezia, Aug 22 2025

Formula

G.f.: (x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)).

a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)^2*(2*cos(r*Pi/10))^(2*n), n >= 1. - Herbert Kociemba, Jun 16 2004

For n > 0, a(n) = (phi^(2*n+1) + 1/phi^(2*n+1))/(2*sqrt(5)) + 5^(n/2-1)*(phi^(n+2) + 1/phi^(n+2))/2, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 22 2025