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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A033192.

Sequence in context: A094817 A033565 A094831 * A071738 A026012 A191993

Adjacent sequences: A033190 A033191 A033192 * A033194 A033195 A033196

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved