0,2

a(n) is the number of permutations of 0..n with each element moved by -1 to 1 places and every 6 consecutive elements having their maximum within 6 of their minimum.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).

Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,1,-2,4,-2,0,2,-2,4,-3,1,0,-2,1,-1).

G.f.: (1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 5*x^6 + 3*x^8 + 2*x^9 - x^10 + x^11 - 4*x^12 - x^13 - 2*x^14 - 2*x^15 - x^17)/(1 - x - x^4 - x^5 + 2*x^6 - 4*x^7 + 2*x^8 - 2*x^10 + 2*x^11 - 4*x^12 + 3*x^13 - x^14 + 2*x^16 - x^17 + x^18).

a(n) = a(n-1) + a(n-4) + a(n-5) - 2*a(n-6) + 4*a(n-7) - 2*a(n-8) + 2*a(n-10) - 2*a(n-11) + 4*a(n-12) - 3*a(n-13) + a(n-14) - 2*a(n-16) + a(n-17) - a(n-18) for n >= 18.

For n = 7, the 29 subsets are {}, {1}, {2}, {3}, {1,3}, {4}, {1,4}, {2,4}, {5}, {1,5}, {2,5}, {3,5}, {1,3,5}, {6}, {1,6}, {2,6}, {3,6}, {1,3,6}, {4,6}, {1,4,6}, {2,4,6}, {7}, {2,7}, {3,7}, {4,7}, {2,4,7}, {5,7}, {2,5,7}, {3,5,7}.

CoefficientList[Series[(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 5*x^6 + 3*x^8 + 2*x^9 - x^10 + x^11 - 4*x^12 - x^13 - 2*x^14 - 2*x^15 - x^17)/(1 - x - x^4 - x^5 + 2*x^6 - 4*x^7 + 2*x^8 - 2*x^10 + 2*x^11 - 4*x^12 + 3*x^13 - x^14 + 2*x^16 - x^17 + x^18), {x, 0, 41}], x]

LinearRecurrence[{1, 0, 0, 1, 1, -2, 4, -2, 0, 2, -2, 4, -3, 1, 0, -2, 1, -1}, {1, 2, 3, 5, 8, 13, 21, 29, 45, 66, 99, 148, 218, 337, 497, 755, 1131, 1699}, 42]

Column k=33 of A376033.

Sequence in context: A080787 A065124 A048817 * A011185 A240733 A283936

Adjacent sequences: A385867 A385868 A385869 * A385871 A385872 A385873

easy,nonn

Michael A. Allen, Jul 11 2025

approved