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

%I A078789 #42 Nov 20 2024 20:44:09
%S A078789 1,3,10,35,127,474,1807,6995,27370,107883,427351,1698458,6765175,
%T A078789 26985675,107746282,430470899,1720537327,6878624730,27505271455,
%U A078789 109996928003,439924466026,1759532283963,7037695641415,28149647662490,112595619434887,450374698997499
%N A078789 Expansion of (1-4*x+2*x^2)/(1-7*x+13*x^2-4*x^3).
%C A078789 Number of walks of length 2*n+1 between two adjacent vertices in the cycle graph C_10. - _Herbert Kociemba_, Jul 02 2004
%C A078789 Also, with offset 1, the cogrowth sequence of the 20-element group D10 = <S,T | S^10, T^2, (ST)^2>. - _Sean A. Irvine_, Nov 20 2024
%H A078789 G. C. Greubel, <a href="/A078789/b078789.txt">Table of n, a(n) for n = 0..1000</a>
%H A078789 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
%H A078789 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-13,4).
%F A078789 G.f.: (1 - 4*x + 2*x^2) / (1 - 7*x + 13*x^2 - 4*x^3).
%F A078789 a(5*n + 3) = A049016(10*n + 3), a(5*n + 4) = A049016(10*n + 5).
%F A078789 From _Herbert Kociemba_, Jul 02 2004: (Start)
%F A078789 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.
%F A078789 a(n) = 7a(n-1)-13a(n-2)+4a(n-3). (End)
%F A078789 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
%F A078789 a(n) = (Lucas(2*n+2) + 2^(2*n+1))/5. - _Greg Dresden_, Jan 26 2023
%t A078789 LinearRecurrence[{7, -13, 4}, {1, 3, 10}, 25] (* _Vincenzo Librandi_, Dec 18 2018 *)
%t A078789 CoefficientList[Series[(1-4x+2x^2)/(1-7x+13x^2-4x^3),{x,0,40}],x] (* _Harvey P. Dale_, Feb 17 2024 *)
%o A078789 (PARI) {a(n) = polcoeff( (1 - 4*x + 2*x^2) / (1 - 7*x + 13*x^2 - 4*x^3) + x * O(x^n), n)}
%o A078789 (PARI) {a(n) = sum( k=-n\5, n\5, binomial(2*n + 1, n+1 + 5*k))}
%o A078789 (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
%Y A078789 Cf. A000032, A001622, A049016.
%K A078789 nonn,easy
%O A078789 0,2
%A A078789 _Michael Somos_, Dec 03 2002