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A187275 a(n) = (n/4)*5^(n/2)*((1+sqrt(5))^2+(-1)^n*(1-sqrt(5))^2). 5

%I

%S 0,5,30,75,300,625,2250,4375,15000,28125,93750,171875,562500,1015625,

%T 3281250,5859375,18750000,33203125,105468750,185546875,585937500,

%U 1025390625,3222656250,5615234375,17578125000,30517578125,95214843750,164794921875,512695312500,885009765625,2746582031250

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

%H G. C. Greubel, <a href="/A187275/b187275.txt">Table of n, a(n) for n = 0..1000</a>

%H R. Kemp, <a href="http://dx.doi.org/10.1016/0012-365X(82)90123-6">On the number of words in the language {w in Sigma* | w = w^R }^2</a>, Discrete Math., 40 (1982), 225-234. See Lemma 1.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-25).

%F a(n) = 10*a(n-2) - 25*a(n-4). - _Colin Barker_, Jul 25 2013

%F G.f.: 5*x*(x+1)*(5*x+1) / (5*x^2-1)^2. - _Colin Barker_, Jul 25 2013

%F a(2*n) = 6*n*5^n, a(2*n+1) = (2*n+1)*5^(n+1). - _Andrew Howroyd_, Mar 28 2016

%p See A187272.

%t LinearRecurrence[{0, 10, 0, -25}, {0, 5, 30, 75}, 30] (* _Vincenzo Librandi_, Mar 29 2016 *)

%o (MAGMA) /* By definition: */ Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); [Integers()!((n/4)*r^n*((1+r)^2+(-1)^n*(1-r)^2)): n in [0..30]]; // _Bruno Berselli_, Mar 29 2016

%o (MAGMA) I:=[0,5,30,75]; [n le 4 select I[n] else 10*Self(n-2)-25*Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Mar 29 2016

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 07 2011

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Last modified July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)