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 A094803 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 1, s(2n) = 3. 5
 1, 3, 9, 28, 90, 296, 988, 3328, 11272, 38304, 130416, 444544, 1516320, 5174144, 17659840, 60282880, 205795456, 702583296, 2398676736, 8189409280, 27960021504, 95460743168, 325921881088, 1112763940864, 3799207806976 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general a(n)= 2/m*Sum_{r=1..m-1} sin(r*j*Pi/m)sin(r*k*Pi/m)(2*cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k. Counts all paths of length (2*n+1), n>=0, starting and ending at the initial node and ending at the nodes 1, 2, 3, 4 and 5 on the path graph P_7, see the Maple program. - Johannes W. Meijer, May 29 2010 LINKS FORMULA a(n)=(1/4)*Sum_{k=1..7} Sin(Pi*k/8)Sin(3Pi*k/8)(2Cos(Pi*k/8))^(2n). a(n)= 6a(n-1)-10a(n-2)+4a(n-3). G.f.: (-1+2x+2x^2)/(4(-1+2x)(1-4x+2x^2)). MAPLE with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=25; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..5); od: seq(a(2*n+1), n=0..nmax-1); # Johannes W. Meijer, May 29 2010 MATHEMATICA f[n_] := FullSimplify[ TrigToExp[(1/4)Sum[ Sin[Pi*k/8]Sin[3Pi*k/8](2Cos[Pi*k/8])^(2n), {k, 1, 7}]]]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *) CROSSREFS Cf. A006012, A030436 and A024175. Sequence in context: A094790 A007822 A094164 * A094826 A033190 A071724 Adjacent sequences:  A094800 A094801 A094802 * A094804 A094805 A094806 KEYWORD nonn AUTHOR Herbert Kociemba, Jun 11 2004 STATUS approved

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Last modified August 26 05:50 EDT 2019. Contains 326330 sequences. (Running on oeis4.)