A288714 - OEIS

%I #22 Feb 16 2025 08:33:48

%S 26,444,3654,22888,124850,628860,3014438,13987152,63462906,283337380,

%T 1249770830,5460869112,23680912034,102049764684,437447065590,

%U 1866647382688,7933717075274,33602668068852,141880252869278,597395676419400,2509073159290866,10514236156062364

%N Number of (undirected) paths on the 2n-crossed prism graph.

%C Sequence extended to n=1 using recurrence. - _Andrew Howroyd_, Jun 19 2017

%H Andrew Howroyd, <a href="/A288714/b288714.txt">Table of n, a(n) for n = 1..200</a>

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

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (16, -105, 366, -732, 840, -512, 128).

%F a(n) = n*(163*4^n-9*2^(n+3)*n-87*2^(n+1)-4)/6.

%F From _Andrew Howroyd_, Jun 19 2017: (Start)

%F a(n) = 16*a(n-1)-105*a(n-2)+366*a(n-3)-732*a(n-4) +840*a(n-5)-512*a(n-6)+128*a(n-7) for n>7.

%F G.f.: 2*x*(13+14*x-360*x^2+764*x^3-580*x^4+152*x^5)/((1-x)^2*(1-2*x)^3*(1-4*x)^2).

%F (End)

%t Table[n (163 4^n - 9 2^(n + 3) n - 87 2^(n + 1) - 4)/6, {n, 20}]

%t LinearRecurrence[{16, -105, 366, -732, 840, -512, 128}, {26, 444, 3654, 22888, 124850, 628860, 3014438}, 20]

%t CoefficientList[Series[-((2 (13 + 14 x - 360 x^2 + 764 x^3 - 580 x^4 + 152 x^5))/((-1 + 2 x)^3 (1 - 5 x + 4 x^2)^2)), {x, 0, 20}], x]

%o (PARI)

%o Vec(2*(13+14*x-360*x^2+764*x^3-580*x^4+152*x^5)/((1-x)^2*(1-2*x)^3*(1-4*x)^2) + O(x^20)) \\ _Andrew Howroyd_, Jun 19 2017

%Y Cf. A234617, A137885.

%K nonn,changed

%O 1,1

%A _Eric W. Weisstein_, Jun 13 2017

%E a(1) prepended and terms a(11) and beyond from _Andrew Howroyd_, Jun 19 2017