A374737 - OEIS

%I #22 Oct 03 2024 15:13:10

%S 1,2,3,5,8,13,18,28,41,62,91,141,208,317,473,719,1069,1623,2425,3666,

%T 5487,8295,12424,18751,28130,42416,63657,95944,144083,217023,326060,

%U 490985,737849,1110753,1669685,2512993,3778125,5685594,8549027,12863637,19344100,29104549

%N Number of subsets of {1,2,...,n} such that no two elements differ by 1 or 5.

%C a(n) is the number of permutations of 0..n with each element moved by -1 to 1 places and every 5 consecutive elements having their maximum within 5 of their minimum.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,-1,1,1,1,0,1).

%F a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) + a(n-6) + a(n-7) + a(n-8) + a(n-10) for n >= 10.

%F G.f.: (1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8 + x^9)/(1 - x^2 - x^3 - x^4 + x^5 - x^6 - x^7 - x^8 - x^10).

%e For n = 6, the 18 subsets are {}, {1}, {2}, {3}, {4}, {5}, {6}, {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6}, {1,3,5}, {2,4,6}.

%t CoefficientList[Series[(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8 + x^9)/(1 - x^2 - x^3 - x^4 + x^5 - x^6 - x^7 - x^8 - x^10),{x,0,41}],x]

%Y See A375981 for other sequences related to restricted combinations.

%Y Column k=17 of A376033.

%K easy,nonn

%O 0,2

%A _Michael A. Allen_, Jul 18 2024