A078789 Expansion of (1-4*x+2*x^2)/(1-7*x+13*x^2-4*x^3).

1, 3, 10, 35, 127, 474, 1807, 6995, 27370, 107883, 427351, 1698458, 6765175, 26985675, 107746282, 430470899, 1720537327, 6878624730, 27505271455, 109996928003, 439924466026, 1759532283963, 7037695641415, 28149647662490, 112595619434887, 450374698997499

Offset: 0

Author

Michael Somos, Dec 03 2002

Keywords

• nonn
• easy

Comments

Number of walks of length 2*n+1 between two adjacent vertices in the cycle graph C_10. - Herbert Kociemba, Jul 02 2004

Also, with offset 1, the cogrowth sequence of the 20-element group D10 = . - Sean A. Irvine, Nov 20 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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 (7,-13,4).

Crossrefs

Cf. A000032, A001622, A049016.

Programs

Magma

I:=[1,3,10]; [n le 3 select I[n] else 7*Self(n-1)-13*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 18 2018

Mathematica

LinearRecurrence[{7, -13, 4}, {1, 3, 10}, 25] (* Vincenzo Librandi, Dec 18 2018 *)
CoefficientList[Series[(1-4x+2x^2)/(1-7x+13x^2-4x^3),{x,0,40}],x] (* Harvey P. Dale, Feb 17 2024 *)

PARI

{a(n) = polcoeff( (1 - 4*x + 2*x^2) / (1 - 7*x + 13*x^2 - 4*x^3) + x * O(x^n), n)}

PARI

{a(n) = sum( k=-n\5, n\5, binomial(2*n + 1, n+1 + 5*k))}

Formula

G.f.: (1 - 4*x + 2*x^2) / (1 - 7*x + 13*x^2 - 4*x^3).

a(5*n + 3) = A049016(10*n + 3), a(5*n + 4) = A049016(10*n + 5).

From Herbert Kociemba, Jul 02 2004: (Start)

a(n) = [4^(n+1) + (sqrt(5)+3)*phi^(2n) - (sqrt(5)-3)*phi^(-2n)]/10, where phi is the golden ratio (1+sqrt(5))/2.

a(n) = 7a(n-1)-13a(n-2)+4a(n-3). (End)

a(n) = Sum_{k= -floor(n/5)..floor(n/5)} binomial(2*n,n+5*k)/2 = ((3+sqrt(5))^n + (3-sqrt(5))^n + 2^(3*n-1))/(5*2^n). - Mircea Merca, Jan 28 2012

a(n) = (Lucas(2*n+2) + 2^(2*n+1))/5. - Greg Dresden, Jan 26 2023