A301654 Circumference of the n-triangular honeycomb acute knight graph.

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, 116, 125, 135, 144, 154, 165, 175, 186, 198, 209, 221, 234, 246, 259, 273, 286, 300, 315, 329, 344, 360, 375, 391, 408, 424, 441, 459, 476, 494, 513, 531, 550, 570, 589, 609, 630, 650

Offset

1,4

Comments

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

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

Links

Table of n, a(n) for n=1..62.

Eric Weisstein's World of Mathematics, Graph Circumference

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

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

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

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

a(n) = floor((n+2)*(n-1)/6). - Stefanos Papanicolopulos, Dec 18 2020

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

Mathematica

Table[(3 n (n + 1) - 14 - 4 Cos[2 n Pi/3] + 4 Sqrt[3] Sin[2 n Pi/3])/18, {n, 20}]
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 1, 3, 4}, 20]
CoefficientList[Series[x^2 (-1 - x + x^2)/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x]

Crossrefs

Sequence in context: A105527 A094345 A243302 * A289233 A039889 A118098

Adjacent sequences: A301651 A301652 A301653 * A301655 A301656 A301657

Keyword

nonn,easy

Author

Eric W. Weisstein, Mar 25 2018

Status

approved