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A091002 Number of walks of length n between non-adjacent nodes on the Petersen graph.

%I A091002 #29 Mar 19 2025 07:22:41
%S A091002 0,0,1,2,9,22,77,210,673,1934,5973,17578,53417,158886,479389,1432706,
%T A091002 4309041,12905278,38759525,116191194,348748345,1045895510,3138385581,
%U A091002 9413758642,28244072129,84726623982,254191056757,762550800650,2287697141193,6863001945094
%N A091002 Number of walks of length n between non-adjacent nodes on the Petersen graph.
%C A091002 Binomial transform of A091005.
%H A091002 G. C. Greubel, <a href="/A091002/b091002.txt">Table of n, a(n) for n = 0..1000</a>
%H A091002 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-6).
%F A091002 3^n = A091000(n) + 3*A091001(n) + 6*a(n).
%F A091002 G.f.: x^2/((1-x)*(1+2*x)*(1-3*x)).
%F A091002 a(n) = (3^(n+1) - (-2)^(n+1) - 5)/30.
%F A091002 a(n) = (A000244(n) - A001045(n+1)*(-1)^n + 4*A001045(n)*(-1)^n)/10.
%F A091002 a(n) = Sum_{k=1..n} binomial(n-k, k)*6^(k-1). - _Zerinvary Lajos_, Sep 30 2006
%F A091002 E.g.f.: (3*exp(3*x) + 2*exp(-2*x) - 5*exp(x))/30. - _G. C. Greubel_, Feb 01 2019
%p A091002 a:=n->sum(binomial(n-k, k)*6^(k-1), k=1..n): seq(a(n),n=0..27); # _Zerinvary Lajos_, Sep 30 2006
%t A091002 Table[(3^n -(-2)^n - 5)/30, {n, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 20 2011 *)
%t A091002 LinearRecurrence[{2,5,-6}, {0,0,1}, 30] (* _G. C. Greubel_, Feb 01 2019 *)
%o A091002 (Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,1,6, lambda n: 1); [next(it) for i in range(0,29)] # _Zerinvary Lajos_, Jul 03 2008
%o A091002 (Sage) [(3^(n+1) - (-2)^(n+1) - 5)/30 for n in range(30)] # _G. C. Greubel_, Feb 01 2019
%o A091002 (PARI) vector(30, n, n--; (3^(n+1) - (-2)^(n+1) - 5)/30) \\ _G. C. Greubel_, Feb 01 2019
%o A091002 (Magma) [(3^(n+1) - (-2)^(n+1) - 5)/30: n in [0..30]]; // _G. C. Greubel_, Feb 01 2019
%o A091002 (GAP) List([0..30], n -> (3^(n+1) - (-2)^(n+1) - 5)/30); # _G. C. Greubel_, Feb 01 2019
%K A091002 easy,nonn
%O A091002 0,4
%A A091002 _Paul Barry_, Dec 12 2003