A224809 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=2, r=4, I={-1,1,2,3}.

1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 24, 36, 54, 81, 117, 169, 247, 361, 532, 784, 1148, 1681, 2460, 3600, 5280, 7744, 11352, 16641, 24381, 35721, 52353, 76729, 112462, 164836, 241570, 354025, 518840, 760384, 1114416, 1633284

Offset: 0

Author

Vladimir Baltic, May 16 2013

Keywords

• nonn

Comments

Number of subsets of {1,2,...,n-4} without differences equal to 2 or 4.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..4096
• Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
• 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,-1,1,1,0,0,-1).

Crossrefs

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

Programs

Mathematica

CoefficientList[Series[-(x-1)*(1+x+x^2)/((x^3+x-1)*(x^6-x^4-1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

PARI

N = 42; x = 'x + O('x^N);
Vec(Ser(-(x-1)*(1+x+x^2)/((x^3+x-1)*(x^6-x^4-1))))  \\ Gheorghe Coserea, Nov 11 2016

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-9).

G.f.: -(x-1)*(1+x+x^2) / ( (x^3+x-1)*(x^6-x^4-1) ).

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