A085282 Expansion of (1 - 5*x + 5*x^2)/((1-x)*(1-3*x)*(1-4*x)).

1, 3, 10, 35, 126, 463, 1730, 6555, 25126, 97223, 379050, 1486675, 5858126, 23166783, 91869970, 365088395, 1453179126, 5791193143, 23100202490, 92207099715, 368247268126, 1471245680303, 5879752544610, 23503319648635, 93966207005126, 375723613252263, 1502470808704330, 6008612301903155, 24030636408870126

Offset: 0

Author

Paul Barry, Jun 25 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A085281.

Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
• Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

Crossrefs

Cf. A085281.

Programs

Magma

[4^n/3+3^n/2+1/6: n in [0..35]]; // Vincenzo Librandi, May 29 2011

Mathematica

CoefficientList[Series[(1 - 5*x + 5*x^2)/((1-x)*(1-3*x)*(1-4*x)), {x, 0, 50}], x] (* Stefano Spezia, Sep 09 2018 *)
Table[(2^(2*n+1) +3^(n+1) +1)/6, {n,0,40}] (* G. C. Greubel, Nov 11 2024 *)
LinearRecurrence[{8,-19,12},{1,3,10},30] (* Harvey P. Dale, Jul 20 2025 *)

PARI

apply( {A085282(n)=(4^n*2+3^(n+1))\/6}, [0..29]) \\ M. F. Hasler, Feb 07 2020

SageMath

def A085282(n): return (2^(2*n+1) +3^(n+1) +1)//6
[A085282(n) for n in range(41)] # G. C. Greubel, Nov 11 2024

Formula

a(n) = 4^n/3 + 3^n/2 + 1/6.

a(n) = Sum_{k=-floor(n/6)..floor(n/6)} binomial(2*n, n+6*k)/2. - Mircea Merca, Jan 28 2012

a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3) for n>2. - Colin Barker, Feb 07 2020

E.x.p.: (1/6)*(exp(x) + 3*exp(3*x) + 2*exp(4*x)). - G. C. Greubel, Nov 11 2024