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A095931 Number of walks of length 2n between two nodes at distance 4 in the cycle graph C_10. 1
1, 7, 36, 165, 715, 3004, 12393, 50559, 204820, 826045, 3321891, 13333932, 53457121, 214146295, 857417220, 3431847189, 13733091643, 54947296924, 219828275865, 879415437615, 3517929664756, 14072420067757, 56291516582931, 225170873858700, 900696081703825 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

In general 2^n/m*Sum_{r=0..m-1} Cos(2Pi*k*r/m)Cos(2Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=10 and k=4.

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..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).

FORMULA

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

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

a(n) = 4^n/(10*Sum_{r=0..9} cos(4*Pi*r/5)*cos(Pi*r/5)^(2*n) ).

From Mircea Merca, Jun 25 2011: (Start)

a(n) = (4^n - (2*cos(Pi/5))^(2*n+1) + (2*cos(2*Pi/5))^(2*n+1))/5.

a(n) = (4^n - ((sqrt(5)+1)/2)^(2*n+1) + ((sqrt(5)-1)/2)^(2*n+1))/5.

a(n) = Sum_{k=1..floor((n+3)/5)} C(2*n+1,n+3-5*k). (End)

5*a(n) = 4^n - A002878(n). - R. J. Mathar, Oct 13 2012

MAPLE

seq(sum(binomial(2*n+1, n+3-5*k), k=1..floor((n+3)*(1/5))), n=2..20)  # Mircea Merca, Jun 25 2011

MATHEMATICA

f[n_]:=FullSimplify[TrigToExp[(4^n/10)Sum[Cos[4Pi*k/5]Cos[Pi*k/5]^(2n), {k, 0, 9}]]]; Table[f[n], {n, 2, 35}]

LinearRecurrence[{7, -13, 4}, {1, 7, 36}, 25] (* Vincenzo Librandi, Dec 20 2018 *)

PROG

(PARI) x='x+O('x^66); /* that many terms */

Vec(x^2/((1-4*x)*(1-3*x+x^2))) /* show terms */ /* Joerg Arndt, Jun 25 2011 */

(GAP) a:=[1, 7, 36];; for n in [4..25] do a[n]:=7*a[n-1]-13*a[n-2]+4*a[n-3]; od; a; # Muniru A Asiru, Dec 19 2018

(MAGMA) I:=[1, 7, 36]; [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 20 2018

CROSSREFS

Sequence in context: A085354 A051198 A003516 * A292486 A026856 A038748

Adjacent sequences:  A095928 A095929 A095930 * A095932 A095933 A095934

KEYWORD

nonn

AUTHOR

Herbert Kociemba, Jul 12 2004

STATUS

approved

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Last modified May 26 00:47 EDT 2022. Contains 354073 sequences. (Running on oeis4.)