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%I #52 Jan 01 2021 12:47:19
%S 0,2,14,78,388,1810,8106,35294,150536,632034,2620870,10759342,
%T 43804812,177105266,711809378,2846259390,11330543632,44929049794,
%U 177540878718,699402223118,2747583822740,10766828545746,42095796462874,164244726238366,639620518118424,2486558615814050,9651161613824822,37403957244654702
%N a(n) = n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2.
%C This was originally based on a graph theory formula in the Wikipedia which turned out to be wrong.
%H Colin Barker, <a href="/A278417/b278417.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-18,8,-1).
%F From _Colin Barker_, Nov 21 2016: (Start)
%F a(n) = 7*a(n-1) - 10*a(n-2) - 10*a(n-3) + 7*a(n-4) - a(n-5) for n>6.
%F G.f.: 2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2.
%F (End)
%p f:=n->expand(n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2); # _N. J. A. Sloane_, May 13 2017
%t Table[Simplify[(n/2) (((2 + #)^n + (2 - #)^n)) &@ Sqrt@ 3], {n, 3, 27}] (* or *)
%t Drop[#, 3] &@ CoefficientList[Series[2 x^3*(39 - 118 x + 55 x^2 - 7 x^3)/(1 - 4 x + x^2)^2, {x, 0, 27}], x] (* _Michael De Vlieger_, Nov 24 2016 *)
%t LinearRecurrence[{8,-18,8,-1},{0,2,14,78},30] (* _Harvey P. Dale_, Jan 01 2021 *)
%o (Python)
%o import math
%o def p(n):
%o m=math.sqrt(3)
%o n=float(n)
%o x=2+m
%o y=2-m
%o return round((n/2)*(x**n+y**n),0)
%o for i in range(3,531):
%o print str(i)+" "+str(int(p(i))) \\_Indranil Ghosh_, Nov 21 2016
%o (PARI) vector(25, n, n+=2; n*((2+sqrt(3))^n + ((2-sqrt(3))^n))/2) \\ _Colin Barker_, Nov 21 2016
%o (PARI) Vec(2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2 + O(x^30)) \\ _Colin Barker_, Nov 21 2016
%Y Cf. A030019, A069996, A139400, A193153.
%K nonn,easy
%O 0,2
%A _Indranil Ghosh_, Nov 21 2016
%E Entry revised by _N. J. A. Sloane_, May 13 2017
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