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Subsets of {1,2,...,n-6} without differences equal to 3 or 6.
5

%I #26 Sep 04 2024 09:55:47

%S 1,1,1,1,1,1,1,2,4,8,12,18,27,36,48,64,96,144,216,324,486,729,1053,

%T 1521,2197,3211,4693,6859,10108,14896,21952,32144,47068,68921,100860,

%U 147600,216000,316800,464640,681472,998976

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

%C 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}.

%H G. C. Greubel, <a href="/A224810/b224810.txt">Table of n, a(n) for n = 0..1000</a>

%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135

%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, 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).

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

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

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

%F 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)

%F 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))

%t 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 *)

%o (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

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

%K nonn,easy

%O 0,8

%A _Vladimir Baltic_, May 16 2013