OFFSET
1,5
COMMENTS
Sequence extended to n=1 and n=2 using the formula/recurrence.
LINKS
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
FORMULA
a(n) = (2*(n - 4)*n*(n^2 - 4*n + 5) - (2*n^2 - 8*n + 9)*(-1)^n + 9)/32.
G.f.: -4*x^5*(1 + x + x^2)/((-1 + x)^5*(1 + x)^3).
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8).
From Eric W. Weisstein, Mar 20 2026: (Start)
a(n) = n*(n-2)^2*(n-4)/16 for n even.
a(n) = (n-1)^2*(n-3)^2/16 for n odd.
a(n) = Z(n,n-2), where Z(m,n)=floor(n/2)*floor((n-1)/2)*floor(m/2)*floor((m-1)/2) is the function in Zarankiewicz's conjecture. (End)
E.g.f.: (x*(x^3 - 2*x^2 + 3*x - 9)*cosh(x) + (x^4 - 2*x^3 + 5*x^2 - 3*x + 9)*sinh(x))/16. - Stefano Spezia, Mar 22 2026
MATHEMATICA
Table[(2 (n - 4) n (n^2 - 4 n + 5) - (2 n^2 - 8 n + 9) (-1)^n + 9)/32, {n, 20}]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 0, 0, 4, 12, 36, 72}, 20]
CoefficientList[Series[-4 x^4 (1 + x + x^2)/((-1 + x)^5 (1 + x)^3), {x, 0, 20}], x]
Table[Piecewise[{{n (n - 2)^2 (n - 4)/16, Mod[n, 2] == 0}, {(n - 1)^2 (n - 3)^2/16, Mod[n, 2] == 1}}], {n, 20}] (* Eric W. Weisstein, Mar 20 2026 *)
Times @@@ Table[Floor[(n - Range[0, 3])/2], {n, 20}] (* Eric W. Weisstein, Mar 20 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 28 2019
STATUS
approved