A387457 - OEIS

%I #14 Sep 03 2025 11:41:28

%S 2772,13096,43400,142680,413168,1162304,3080736,7966880,19932352,

%T 48943488,117719680,279180160,652834560,1510279168,3458421248,

%U 7853658624,17697041408,39614228480,88136509440,195043481600,429506424832,941650132992,2056105369600,4472971829248

%N Number of Hamiltonian paths in the n-Goldberg graph.

%H Andrew Howroyd, <a href="/A387457/b387457.txt">Table of n, a(n) for n = 3..1000</a>

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,-16,44,-8,-48,32).

%F G.f.: x^3*(2772 - 3536*x - 13000*x^2 + 31400*x^3 - 8144*x^4 - 34912*x^5 + 23360*x^6 + 768*x^7)/((1 - 2*x)^3*(1 - 2*x^2)^2). - _Andrew Howroyd_, Aug 29 2025

%F a(n) = 6*a(n-1)-8*a(n-2)-16*a(n-3)+44*a(n-4)-8*a(n-5)-48*a(n-6)+32*a(n-7) for n > 8. - _Eric W. Weisstein_, Sep 03 2025

%t Join[{2772}, Table[2^((n - 5)/2) (1124 (1 - (-1)^n) + 929 Sqrt[2] (1 + (-1)^n)) n + 3 2^(n - 1) n (75 n - 241), {n, 4, 20}]] // Expand

%t Join[{2772}, LinearRecurrence[{6, -8, -16, 44, -8, -48, 32}, {13096, 43400, 142680, 413168, 1162304, 3080736, 7966880}, 20]]

%t CoefficientList[Series[-4 (693 - 884 x - 3250 x^2 + 7850 x^3 - 2036 x^4 - 8728 x^5 + 5840 x^6 + 192 x^7)/((-1 + 2 x)^3 (-1 + 2 x^2)^2), {x, 0, 20}], x]

%K nonn

%O 3,1

%A _Eric W. Weisstein_, Aug 29 2025

%E a(8) onwards from _Andrew Howroyd_, Aug 29 2025