A376033 -id:A376033 - cp's OEIS frontend

A375186 Number of subsets of {1,2,...,n} such that no two elements differ by 1, 2, 4, or 5.

1, 2, 3, 4, 6, 8, 10, 14, 19, 25, 35, 48, 64, 88, 120, 161, 220, 300, 405, 552, 752, 1018, 1385, 1885, 2556, 3475, 4727, 6416, 8720, 11857, 16102, 21881, 29745, 40406, 54905, 74626, 101389, 137769, 187235, 254404, 345689, 469781, 638339

Offset: 0

Michael A. Allen, Aug 02 2024

a(n-4) for n>3 is the number of equivalence classes of binary words of length n for the subword 100110 (see A317669 for further explanation).

a(n) is the number of compositions of n+5 into parts 1, 6, 9, 12, 15, 18, ...

			For n = 6, the 10 subsets are {}, {1}, {2}, {3}, {4}, {1,4}, {5}, {2,5}, {6}, {3,6}.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1).
Index entries for subsets of {1,..,n} with disallowed differences

Column k=27 of A376033.

CoefficientList[Series[(1 + x + x^2 + x^4 + x^5)/(1 - x - x^3 + x^4 - x^6),{x,0,42}],x]
LinearRecurrence[{1, 0, 1, -1, 0, 1}, {1, 2, 3, 4, 6, 8}, 42]

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) for n>= 6.

G.f.: (1 + x + x^2 + x^4 + x^5)/(1 - x - x^3 + x^4 - x^6).

A375979 Number of subsets of {1,2,...,n} such that no two elements differ by 4 or 5.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192, 314, 504, 767, 1120, 1612, 2324, 3412, 5148, 7900, 12169, 18631, 28152, 42024, 62364, 92576, 138141, 207629, 313718, 474796, 717456, 1080320, 1620994, 2427447, 3634800, 5450293, 8188936, 12323172, 18555880, 27930853

Offset: 0

Michael A. Allen, Sep 20 2024

			For n = 6, the 32 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, {5}, {2,5}, {3,5}, {2,3,5}, {4,5}, {2,4,5}, {3,4,5}, {2,3,4,5}, {6}, {3,6}, {4,6}, {3,4,6}, {5,6}, {3,5,6}, {4,5,6}, {3,4,5,6}.

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 (1,0,1,-1,0,1,1,2,3,-1,-2,-3).
Index entries for subsets of {1,..,n} with disallowed differences

Column k=24 of A376033.

CoefficientList[Series[(1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 6*x^5 + 3*x^6 - x^7 - 3*x^8 - 6*x^9 - 5*x^10 - 3*x^11)/(1 - x - x^3 + x^4 - x^6 - x^7 - 2*x^8 - 3*x^9 + x^10 + 2*x^11 + 3*x^12),{x,0,38}],x]
LinearRecurrence[{1, 0, 1, -1, 0, 1, 1, 2, 3, -1, -2, -3}, {1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192}, 39]

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) + a(n-7) + 2*a(n-8) + 3*a(n-9) - a(n-10) - 2*a(n-11) - 3*a(n-12) for n >= 12.

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

A375984 Number of subsets of {1,2,...,n} such that no two elements differ by 3, 4, or 5.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 25, 33, 49, 77, 121, 181, 258, 356, 488, 680, 976, 1432, 2113, 3089, 4449, 6329, 8961, 12729, 18226, 26292, 38056, 55012, 79200, 113548, 162425, 232401, 333201, 478853, 689177, 991949, 1426322, 2048244, 2938696, 4215552, 6049984, 8688816

Offset: 0

Michael A. Allen, Sep 21 2024

			For n = 6, the 20 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {2,4}, {3,4}, {2,3,4}, {5}, {3,5}, {4,5}, {3,4,5}, {6}, {4,6}, {5,6}, {4,5,6}.

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 (1,0,0,0,0,1,1,2).
Index entries for subsets of {1,..,n} with disallowed differences

Column k=28 of A376033.

CoefficientList[Series[(1 + x + 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 3*x^6 + 2*x^7)/(1 - x - x^6 - x^7 - 2*x^8),{x,0,40}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 1, 1, 2}, {1, 2, 4, 8, 12, 16, 20, 25}, 41]

a(n) = a(n-1) + a(n-6) + a(n-7) + 2*a(n-8) for n >= 8.

G.f.: (1 + x + 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 3*x^6 + 2*x^7)/((1 + x)(1 + x^2)(1 - 2*x + x^2 + x^4 - 2*x^5)).

A375985 Number of subsets of {1,2,...,n} such that no two elements differ by 1, 3, 4, or 5.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 14, 18, 25, 35, 49, 67, 90, 119, 158, 211, 285, 387, 526, 712, 960, 1290, 1733, 2331, 3142, 4241, 5727, 7729, 10422, 14043, 18918, 25490, 34359, 46329, 62478, 84250, 113590, 153123, 206400, 278219, 375056, 505635, 681703, 919076, 1239066

Offset: 0

Michael A. Allen, Sep 21 2024

a(n) is the number of compositions of n+5 into parts 1, 6, and 8.

			For n = 6, the 11 subsets are {}, {1}, {2}, {3}, {1,3}, {4}, {2,4}, {5}, {3,5}, {6}, {4,6}.

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 (1,0,0,0,0,1,0,1).
Index entries for subsets of {1,..,n} with disallowed differences

Column k=29 of A376033.

CoefficientList[Series[(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6 + x^7)/(1 - x - x^6 - x^8),{x,0,43}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 1, 0, 1}, {1, 2, 3, 5, 7, 9, 11, 14}, 44]

a(n) = a(n-1) + a(n-6) + a(n-8) for n >= 8.

G.f.: (1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6 + x^7)/((1 + x)(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + x^6 - x^7)).

A385870 Number of subsets of {1,2,...,n} such that no two elements differ by 1 or 6.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 21, 29, 45, 66, 99, 148, 218, 337, 497, 755, 1131, 1699, 2571, 3824, 5794, 8661, 13041, 19601, 29376, 44311, 66349, 99936, 150000, 225387, 339000, 508631, 765392, 1148865, 1727249, 2595270, 3898324, 5861084, 8801690, 13231745, 19877092, 29869125

Offset: 0

Michael A. Allen, Jul 11 2025

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.

			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}.

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,-1,1,-1,0,1,-1,1,-1,1,0,-1,1,-1).
Index entries for subsets of {1,..,n} with disallowed differences

Column k=33 of A376033.

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]

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.

cp's OEIS Frontend

A375186 Number of subsets of {1,2,...,n} such that no two elements differ by 1, 2, 4, or 5.

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A375979 Number of subsets of {1,2,...,n} such that no two elements differ by 4 or 5.

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A375984 Number of subsets of {1,2,...,n} such that no two elements differ by 3, 4, or 5.

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A375985 Number of subsets of {1,2,...,n} such that no two elements differ by 1, 3, 4, or 5.

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A385870 Number of subsets of {1,2,...,n} such that no two elements differ by 1 or 6.

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