Michael A. Allen - cp's OEIS frontend

A387021 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,7} for all i=1,...,n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773, 3670, 4861, 6388, 8344, 10848, 14019, 18166, 23479, 30556, 39762, 52049, 68125, 89345, 117034, 153078, 199979, 260572, 339546, 441669, 575341

Offset: 0

Michael A. Allen, Aug 13 2025

			a(9)=2: 123456789, 891234567.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

V. Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,4,-3,3,-2,2,-1,1,0,0,-6,3,-6,2,-3,0,0,0,0,4,-1,3,0,0,0,0,0,0,-1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,...,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36),{x,0,55}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 4, -3, 3, -2, 2, -1, 1, 0, 0, -6, 3, -6, 2, -3, 0, 0, 0, 0, 4, -1, 3, 0, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773}, 56]

a(n) = a(n-1) + 4*a(n-9) - 3*a(n-10) + 3*a(n-11) - 2*a(n-12) + 2*a(n-13) - a(n-14) + a(n-15) - 6*a(n-18) + 3*a(n-19) - 6*a(n-20) + 2*a(n-21) - 3*a(n-22) + 4*a(n-27) - a(n-28) + 3*a(n-29) - a(n-36) for n >= 36.

G.f.: (1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36).

A387020 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,5} for all i=1,...,n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130, 175, 231, 305, 400, 540, 729, 999, 1363, 1855, 2510, 3370, 4531, 6070, 8180, 11026, 14921, 20197, 27322, 36940, 49820, 67204, 90528, 122091, 164686, 222344, 300316, 405574, 547768, 739291, 997794, 1346130

Offset: 0

Michael A. Allen, Aug 13 2025

			a(7) = 2: 1234567, 6712345.
a(8) = 3: 12345678, 17823456, 67123458.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltić, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,3,-2,2,-1,1,0,0,-3,1,-2,0,0,0,0,1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,..,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 2*x^7 - x^9 + x^14)/(1 - x - 3*x^7 + 2*x^8 - 2*x^9 + x^10 - x^11 + 3*x^14 - x^15 + 2*x^16 - x^21),{x,0,51}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 3, -2, 2, -1, 1, 0, 0, -3, 1, -2, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130}, 52]

a(n) = a(n-1) + 3*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-10) + a(n-11) - 3*a(n-14) + a(n-15) - 2*a(n-16) + a(n-21) for n >= 21.

G.f.: (1 - 2*x^7 - x^9 + x^14)/((1 - x)*(1 - x + x^2 - 2*x^3 + x^4 - x^5 - x^7 + x^10)*(1 + x + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + x^8 + x^9 + x^10)).

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.

A378005 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,4,5} for all i=1,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 6, 6, 1, 0, 0, 1, 10, 20, 10, 1, 1, 1, 15, 50, 50, 15, 6, 7, 21, 105, 175, 105, 36, 42, 49, 196, 490, 490, 231, 183, 217, 392, 1176, 1764, 1246, 785, 946, 1141, 2646, 5292, 5418, 3613, 3664, 4390, 6601, 14112, 19614

Offset: 0

Michael A. Allen, Nov 13 2024

			a(6) = 1: 561234.
a(7) = 1: 6712345.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,2,3,0,-2,-2,0,-1,-2,-3,1,1,1,0,0,0,1).

See A376743 for other sequences related to strongly restricted permutations.

Cf. A377715, A001263.

CoefficientList[Series[1/((1-x^7-x^6-x^13/(1-x^7-x^6/(1-x^7/(1-x^3))))), {x, 0, 65}],x]

a(n) = a(n-3) + 2*a(n-6) + 3*a(n-7) - 2*a(n-9) - 2*a(n-10) - a(n-12) - 2a(n-13) - 3*a(n-14) + a(n-15) + a(n-16) + a(n-17) + a(n-21) for n >= 21.

G.f.: (1 - x)*(1 + x + x^2 - x^6 - 3*x^7 - 3*x^8 - 2*x^9 - x^10 - x^11 - x^12 - x^13)/(1 - x^3 - 2*x^6 - 3*x^7 + 2*x^9 + 2*x^10 + x^12 + 2*x^13 + 3*x^14 - x^15 - x^16 - x^17 - x^21).

A377715 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,3,4} for all i=1,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 1, 0, 0, 1, 6, 6, 1, 1, 1, 10, 20, 10, 5, 6, 15, 50, 50, 25, 30, 36, 105, 175, 125, 115, 141, 231, 490, 525, 435, 541, 673, 1246, 1820, 1695, 1901, 2361, 3361, 5501, 6301, 6601, 8295, 10440, 15820, 21410, 23286

Offset: 0

Michael A. Allen, Nov 04 2024

			a(5) = 1: 45123.
a(6) = 1: 561234.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,2,2,0,-1,-2,-1,-1,-1,0,0,1).

See A376743 for other sequences related to strongly restricted permutations.

CoefficientList[Series[(1 - x^3 - x^5 - x^6 + x^9)/(1 - x^3 - 2*x^5 - 2*x^6 + x^8 + 2*x^9 + x^10 + x^11 + x^12 - x^15),{x,0,56}],x]
LinearRecurrence[{0, 0, 1, 0, 2, 2, 0, -1, -2, -1, -1, -1, 0, 0, 1}, {1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 1, 0, 0}, 57]

a(n) = a(n-3) + 2*a(n-5) + 2*a(n-6) - a(n-8) - 2*a(n-9) - a(n-10) - a(n-11) - a(n-12) + a(n-15) for n >= 15.

G.f.: (1 - x^3 - x^5 - x^6 + x^9)/(1 - x^3 - 2*x^5 - 2*x^6 + x^8 + 2*x^9 + x^10 + x^11 + x^12 - x^15).

A376743 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,-1,4} for all i=1,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8, 11, 15, 25, 35, 46, 61, 85, 125, 175, 245, 341, 470, 650, 925, 1300, 1810, 2521, 3520, 4915, 6880, 9640, 13476, 18801, 26251, 36721, 51346, 71776, 100335, 140210, 195886, 273813, 382821, 535105, 747850, 1045220

Offset: 0

Michael A. Allen, Oct 03 2024

Other sequences related to strongly restricted permutations pi(i) of i in {1,..,n} along with the sets of allowed p(i)-i (containing at least 3 elements): A000045 {-1,0,1}, A189593 {-1,0,2,3,4,5,6}, A189600 {-1,0,2,3,4,5,6,7}, A006498 {-2,0,2}, A080013 {-2,1,2}, A080014 {-2,0,1,2}, A033305 {-2,-1,1,2}, A002524 {-2,-1,0,1,2}, A080000 {-2,0,3}, A080001 {-2,1,3}, A080004 {-2,0,1,3}, A080002 {-2,2,3}, A080005 {-2,0,2,3}, A080008 {-2,1,2,3}, A080011 {-2,0,1,2,3}, A079999 {-2,-1,3}, A080003 {-2,-1,0,3}, A080006 {-2,-1,1,3}, A080009 {-2,-1,0,1,3}, A080007 {-2,-1,2,3}, A080010 {-2,-1,0,2,3}, A080012 {-2,-1,1,2,3}, A072827 {-2,-1,0,1,2,3}, A224809 {-2,0,4}, A189585 {-2,0,1,3,4}, A189581 {-2,-1,0,3,4}, A072850 {-2,-1,0,1,2,3,4}, A189587 {-2,0,1,3,4,5}, A189588 {-2,-1,0,3,4,5}, A189594 {-2,0,1,3,4,5,6}, A189595 {-2,-1,0,3,4,5,6}, A189601 {-2,0,1,3,4,5,6,7}, A189602 {-2,-1,0,3,4,5,6,7}, A224811 {-2,0,8}, A224812 {-2,0,10}, A224813 {-2,0,12}, A006500 {-3,0,3}, A079981 {-3,1,3}, A079983 {-3,0,1,3}, A079982 {-3,2,3}, A079984 {-3,0,2,3}, A079988 {-3,1,2,3}, A079989 {-3,0,1,2,3}, A079986 {-3,-1,1,3}, A079992 {-3,-1,0,1,3}, A079987 {-3,-1,2,3}, A079990 {-3,-1,0,2,3}, A079993 {-3,-1,1,2,3}, A079985 {-3,-2,2,3}, A079991 {-3,-2,0,2,3}, A079996 {-3,-2,0,1,2,3}, A079994 {-3,-2,1,2,3}, A079997 {-3,-2, -1,1,2,3}, A002526 {-3,-2,-1,0,1,2,3}, A189586 {-3,0,1,2,4}, A189583 {-3,-1,0,2,4}, A189582 {-3,-2,0,1,4}, A189584 {-3,-2,-1,0,4}, A189589 {-3,0,1,2,4,5}, A189590 {-3,-1,0,2,4,5}, A189591 {-3,-2,1,4,5}, A189592 {-3,-2,-1,0,4,5}, A224810 {-3,0,6}, A189596 {-3,0,1,2,4,5,6}, A189597 {-3,-1,0,2,4,5,6}, A189598 {-3,-2,0,1,4,5,6}, A189599 {-3,-2,-1,0,4,5,6}, A224814 {-3,0,9}, A031923 {-4,0,4}, A072856 {-4,-3, -2,-1,0,1,2,3,4}, A224815 {-4,0,8}, A154654 {-5,-4,-3,-2,-1,0,1,2,3,4,5}, A154655 {-6,-5,-4,-3, -2,-1,0,1,2,3,4,5,6}.

[Keyword "less", because this comment should be moved to the Index to the OEIS, it is not appropriate here. - N. J. A. Sloane, Oct 25 2024]

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,2,1,0,-2,-2,0,-1,0,0,1).

See comments for other sequences related to strongly restricted permutations.

CoefficientList[Series[(1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15),{x,0,49}],x]
LinearRecurrence[{0, 0, 1, 1, 1, 2, 1, 0, -2, -2, 0, -1, 0, 0, 1}, {1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8}, 50]

a(n) = a(n-3) + a(n-4) + a(n-5) + 2*a(n-6) + a(n-7) - 2*a(n-9) - 2*a(n-10) - a(n-12) + a(n-15).

G.f.: (1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15).

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)).

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)).

A375980 Number of subsets of {1,2,...,n} such that no two elements differ by 1, 2, or 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 17, 25, 35, 49, 71, 101, 142, 203, 290, 410, 583, 832, 1181, 1677, 2389, 3397, 4825, 6865, 9766, 13879, 19736, 28074, 39913, 56748, 80709, 114765, 163175, 232045, 329975, 469189, 667178, 948743, 1349062, 1918310, 2727839, 3878912, 5515657

Offset: 0

Michael A. Allen, Sep 21 2024

			For n = 6, the 12 subsets are {}, {1}, {2}, {3}, {4}, {1,4}, {5}, {1,5}, {2,5}, {6}, {2,6}, {3,6}.

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

Column k=19 of A376033.

CoefficientList[Series[(1 + x + x^2 + x^5)/(1 - x - x^3 + x^5 - x^6),{x,0,42}],x]
LinearRecurrence[{1, 0, 1, 0, -1, 1}, {1, 2, 3, 4, 6, 9}, 43]

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) for n >= 6.

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

A375977 Number of subsets of {1,2,...,n} such that no two elements differ by 2 or 5.

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 21, 29, 45, 69, 100, 152, 236, 349, 517, 789, 1185, 1757, 2653, 4014, 5992, 8986, 13573, 20363, 30485, 45901, 69041, 103481, 155468, 233908, 351104, 527033, 792405, 1190493, 1787129, 2685209, 4035261, 6059758, 9101828, 13676670, 20544125

Offset: 0

Michael A. Allen, Sep 20 2024

			For n = 6, the 21 subsets are {}, {1}, {2}, {1,2}, {3}, {2,3}, {4}, {1,4}, {3,4}, {5}, {1,5}, {2,5}, {1,2,5}, {4,5}, {1,4,5}, {6}, {2,6}, {3,6}, {2,3,6}, {5,6}, {2,5,6}.

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

Column k=18 of A376033.

CoefficientList[Series[(1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 - x^7 - x^10)/(1 - x - x^3 + x^5 - x^6 - 2*x^7 + x^8 - x^10 + x^11),{x,0,39}],x]
LinearRecurrence[{1, 0, 1, 0, -1, 1, 2, -1, 0, 1, -1}, {1, 2, 4, 6, 9, 15, 21, 29, 45, 69, 100}, 39]

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) + 2*a(n-7) - a(n-8) + a(n-10) - a(n-11) for n >= 11.

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

cp's OEIS Frontend

User: Michael A. Allen

A387021 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,7} for all i=1,...,n.

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A387020 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,5} for all i=1,...,n.

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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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A378005 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,4,5} for all i=1,...,n.

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A377715 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,3,4} for all i=1,...,n.

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A376743 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,-1,4} for all i=1,...,n.

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