OFFSET
1,6
COMMENTS
Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).
FORMULA
a(n) = (S2(n,k) - A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.
a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020
EXAMPLE
For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.
MATHEMATICA
k=5; Table[(StirlingS2[n, k] - If[EvenQ[n], 3StirlingS2[n/2+2, 5] - 11StirlingS2[n/2+1, 5] + 6StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3StirlingS2[(n+3)/2, 5]])/2, {n, 30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)
k = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]
PROG
(PARI) m=30; v=concat([0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018
(Magma) I:=[0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 14 2018
STATUS
approved