OFFSET
1,2
COMMENTS
Also, the maximum detour index of any bipartite graph on n^3 nodes. - Andrew Howroyd, Dec 21 2017
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Detour Index
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,4,-20,0,56,-28,-84,70,70,-84,-28,56,0,-20,4,3,-1).
FORMULA
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
From Colin Barker, Dec 21 2017: (Start)
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).
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.
(End)
a(n) = A296819(n^3). - Andrew Howroyd, Dec 23 2017
MATHEMATICA
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}]
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]
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]
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Dec 20 2017
EXTENSIONS
Terms a(4) and beyond from Andrew Howroyd, Dec 21 2017
STATUS
approved