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A301654
Circumference of the n-triangular honeycomb acute knight graph.
1
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
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
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 25 2018
STATUS
approved