A289143 Matching number of the n-triangular honeycomb acute knight graph.

0, 0, 3, 3, 6, 9, 12, 18, 21, 27, 33, 39, 45, 51, 60, 67, 75, 84, 94, 105, 114, 126, 138, 150, 162, 174, 189, 202, 216, 231, 247, 264, 279, 297, 315, 333, 351, 369, 390, 409, 429, 450, 472, 495, 516, 540, 564, 588, 612, 636, 663, 688, 714, 741, 769, 798, 825, 855, 885, 915, 945, 975, 1008, 1039, 1071, 1104, 1138, 1173, 1206, 1242, 1278, 1314

Offset: 1

Author

Eric W. Weisstein, Jun 26 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Matching Number
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,-1,3,-3,1).

Programs

Mathematica

Table[Piecewise[{{3, n == 4}, {12, n == 7}}, (3 (n^2 + n - 3) + 5 Cos[n Pi/2] + 4 (-1)^n (Cos[n Pi/3] + Sqrt[3] Sin[n Pi/3]) (Cos[n Pi/2] - Sin[n Pi/2]) - 5 Sin[n Pi/2])/12], {n, 50}]

PARI

concat(vector(2), Vec( x^3*(3 - 6*x + 6*x^2 - 3*x^3 + 3*x^5 - 3*x^6 + 3*x^8 - 3*x^9 + 3*x^11 - 3*x^12 + x^13) / ((1 - x)^3*(1 + x^2)*(1 - x^2 + x^4)) + O(x^80))) \\ Colin Barker, Jun 26 2017

Formula

For n > 13, a(n) = (n^2-n+6-2*a(n-6))/2.

From Colin Barker, Jun 26 2017: (Start)

G.f.: x^3*(3 - 6*x + 6*x^2 - 3*x^3 + 3*x^5 - 3*x^6 + 3*x^8 - 3*x^9 + 3*x^11 - 3*x^12 + x^13) / ((1 - x)^3*(1 + x^2)*(1 - x^2 + x^4)).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9) for n>9.

(End)