A301654 - OEIS

%I #14 Dec 27 2020 17:00:07

%S 0,0,1,3,4,6,9,11,14,18,21,25,30,34,39,45,50,56,63,69,76,84,91,99,108,

%T 116,125,135,144,154,165,175,186,198,209,221,234,246,259,273,286,300,

%U 315,329,344,360,375,391,408,424,441,459,476,494,513,531,550,570,589,609,630,650

%N Circumference of the n-triangular honeycomb acute knight graph.

%C Sequence extended to a(1)-a(3) using the formula.

%C a(n) agrees with A243302(n-2) for 3 <= n <= 12.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCircumference.html">Graph Circumference</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F a(n) = (3*n*(n + 1) - 14 - 4*cos(2*n*Pi/3) + 4*sqrt(3)*sin(2*n*Pi/3))/18.

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 5.

%F G.f.: x^3*(-1 - x + x^2)/((-1 + x)^3*(1 + x + x^2)).

%F a(n) = floor((n+2)*(n-1)/6). - _Stefanos Papanicolopulos_, Dec 18 2020

%F E.g.f.: 1 + exp(-x/2)*(exp(3*x/2)*(3*x^2 + 6*x - 14) - 4*cos(sqrt(3)*x/2) + 4*sqrt(3)*sin(sqrt(3)*x/2))/18. - _Stefano Spezia_, Dec 18 2020

%t Table[(3 n (n + 1) - 14 - 4 Cos[2 n Pi/3] + 4 Sqrt[3] Sin[2 n Pi/3])/18, {n, 20}]

%t LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 1, 3, 4}, 20]

%t CoefficientList[Series[x^2 (-1 - x + x^2)/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x]

%K nonn,easy

%O 1,4

%A _Eric W. Weisstein_, Mar 25 2018