A224810 Subsets of {1,2,...,n-6} without differences equal to 3 or 6.

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 96, 144, 216, 324, 486, 729, 1053, 1521, 2197, 3211, 4693, 6859, 10108, 14896, 21952, 32144, 47068, 68921, 100860, 147600, 216000, 316800, 464640, 681472, 998976

Offset: 0

Author

Vladimir Baltic, May 16 2013

Keywords

• nonn
• easy

Comments

Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i not in the set I, i=1..n, with k=3, r=6, I={-2,-1,1,2,3,4,5}.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
• Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 2, 0, -2, 4, 0, 2, 2, 0, 4, -2, 0, 2, -4, 0, -2, -2, 0, -1, -1, 0, -1).

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694,A224808.

Programs

Mathematica

CoefficientList[Series[(1 + x^3 - x^4 - x^5 + x^6 - 2*x^7 - x^8 - x^9 - 2*x^10 - x^12 - x^13 - x^15)/((1 - x)*(1 + x + x^2)*(1 - x - x^3)*(1 + 3*x^3 + 7*x^6 + 9*x^9 + 7*x^12 + 3*x^15 + x^18)), {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2017 *)

PARI

x='x+O('x^50); Vec((1 + x^3 - x^4 - x^5 + x^6 - 2*x^7 - x^8 - x^9 - 2*x^10 - x^12 - x^13 - x^15)/((1 - x)*(1 + x + x^2)*(1 - x - x^3)*(1 + 3*x^3 + 7*x^6 + 9*x^9 + 7*x^12 + 3*x^15 + x^18))) \\ G. C. Greubel, Apr 30 2017

Formula

a(3*k) = (A000930(k))^3.

a(3*k+1) = (A000930(k))^2 * A000930(k+1).

a(3*k+2) = A000930(k) * (A000930(k+1))^2.

a(n) = a(n-1) -a(n-3) +2*a(n-4) -2*a(n-6) +4*a(n-7) +2*a(n-9) +2*a(n-10) +4*a(n-12) -2*a(n-13) +2*a(n-15) -4*a(n-16) -2*a(n-18) -2*a(n-19) -a(n-21) -a(n-22) -a(n-24)

G.f.: (1+x^3-x^4-x^5+x^6-2*x^7-x^8-x^9-2*x^10-x^12-x^13-x^15) / ((1-x)*(1+x+x^2)*(1-x-x^3)*(1+3*x^3+7*x^6+9*x^9+7*x^12+3*x^15+x^18))