A296780 - OEIS

%I #16 Dec 24 2017 10:35:30

%S 0,184,8788,126016,953312,4980744,20000844,66781696,192914176,

%T 498751000,1176318500,2576159424,5295012768,10321114216,19204598812,

%U 34338770944,59257739264,99137135736,161273290356,255920008000,397011388000,603492894664,900354295148

%N Detour index of the n X n X n grid graph.

%C Also, the maximum detour index of any bipartite graph on n^3 nodes. - _Andrew Howroyd_, Dec 21 2017

%H Colin Barker, <a href="/A296780/b296780.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>

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

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,-20,0,56,-28,-84,70,70,-84,-28,56,0,-20,4,3,-1).

%F a(2n-1) = 4*(n - 1)^3*(4*n^2 - 2*n + 1)^3, a(2n) = 8*n^3*(32*n^6 - 10*n^3 + 1). - _Andrew Howroyd_, Dec 21 2017

%F From _Colin Barker_, Dec 21 2017: (Start)

%F G.f.: 4*x^2*(46 + 2059*x + 24729*x^2 + 135948*x^3 + 448126*x^4 + 938845*x^5 + 1358863*x^6 + 1346304*x^7 + 941714*x^8 + 442773*x^9 + 138599*x^10 + 25508*x^11 + 2482*x^12 + 83*x^13 + x^14) / ((1 - x)^10*(1 + x)^7).

%F a(n) = 3*a(n-1) + 4*a(n-2) - 20*a(n-3) + 56*a(n-5) - 28*a(n-6) - 84*a(n-7) + 70*a(n-8) + 70*a(n-9) - 84*a(n-10) - 28*a(n-11) + 56*a(n-12) - 20*a(n-14) + 4*a(n-15) + 3*a(n-16) - a(n-17) for n>17.

%F (End)

%F a(n) = A296819(n^3). - _Andrew Howroyd_, Dec 23 2017

%t Table[Piecewise[{{n^3 (n^6 - 5 n^3/2 + 2)/2, Mod[n, 2] == 0}, {(n - 1)^3 (n^2 + n + 1)^3/2, Mod[n, 2] == 1}}], {n, 20}]

%t LinearRecurrence[{3, 4, -20, 0, 56, -28, -84, 70, 70, -84, -28, 56, 0, -20, 4, 3, -1}, {0, 184, 8788, 126016, 953312, 4980744, 20000844, 66781696, 192914176, 498751000, 1176318500, 2576159424, 5295012768, 10321114216, 19204598812, 34338770944, 59257739264}, 20]

%t CoefficientList[Series[4 x (46 + 2059 x + 24729 x^2 + 135948 x^3 + 448126 x^4 + 938845 x^5 + 1358863 x^6 + 1346304 x^7 + 941714 x^8 + 442773 x^9 + 138599 x^10 + 25508 x^11 + 2482 x^12 + 83 x^13 + x^14)/((1 - x)^10 (1 + x)^7), {x, 0, 20}], x]

%o (PARI) a(n) = if(n%2, (n-1)^3*(n^2 + n + 1)^3, n^3*(n^6 - 5*n^3/2 + 2))/2; \\ _Andrew Howroyd_, Dec 21 2017

%o (PARI) concat(0, Vec(4*x^2*(46 + 2059*x + 24729*x^2 + 135948*x^3 + 448126*x^4 + 938845*x^5 + 1358863*x^6 + 1346304*x^7 + 941714*x^8 + 442773*x^9 + 138599*x^10 + 25508*x^11 + 2482*x^12 + 83*x^13 + x^14) / ((1 - x)^10*(1 + x)^7) + O(x^40))) \\ _Colin Barker_, Dec 21 2017

%Y Cf. A296779, A296819.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Dec 20 2017

%E Terms a(4) and beyond from _Andrew Howroyd_, Dec 21 2017