cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Michael A. Allen, Oct 03 2024

Keywords

Comments

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]

References

  • 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.

Links

Crossrefs

See comments for other sequences related to strongly restricted permutations.

Programs

  • Mathematica
    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]

Formula

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).

A224814 Number of subsets of {1,2,...,n-9} without differences equal to 3, 6 or 9.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 80, 100, 125, 175, 245, 343, 490, 700, 1000, 1400, 1960, 2744, 3724, 5054, 6859, 9386, 12844, 17576, 24336, 33696, 46656, 64800, 90000, 125000, 172500, 238050, 328509, 452295, 622725, 857375, 1182275, 1630295, 2248091, 3106141, 4291691, 5929741, 8190250, 11312500, 15625000, 21562500
Offset: 0

Views

Author

Vladimir Baltic, May 18 2013

Keywords

Comments

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=3, r=9, I={-3,0,9}.

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^4 - x^5 - x^7 - x^8 + 2*x^9 - x^10 - 3*x^12 - x^13 - 2*x^15 + 3*x^16 + 3*x^17 + 2*x^18 - x^20 - 4*x^21 + x^23 + 3*x^24 + 3*x^25 + x^27 - 4*x^28 - x^29 - 2*x^30 + x^31 + 2*x^33 +x^34 - x^36 - x^37 + x^40)/((1 - x - x^4)*(1 - x^9 - x^12)*(1 + x^6 + 4*x^9 - 4*x^12 - 2*x^15 + 4*x^18 - 3*x^21 - 3*x^24 + 7*x^27 - 6*x^30 + 3*x^33 - x^36)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)
  • PARI
    x='x+O('x^50); Vec((1 -x^4 -x^5 -x^7 -x^8 +2*x^9 -x^10 -3*x^12 -x^13 -2*x^15 +3*x^16 +3*x^17 +2*x^18 -x^20 -4*x^21 +x^23 +3*x^24 +3*x^25 +x^27 -4*x^28 -x^29 -2*x^30 +x^31 +2*x^33 +x^34 -x^36 -x^37 +x^40 )/((1-x-x^4)*(1-x^9-x^12)*(1 +x^6 +4*x^9 -4*x^12 -2*x^15 +4*x^18 -3*x^21 -3*x^24 +7*x^27 -6*x^30 +3*x^33 -x^36))) \\ G. C. Greubel, Oct 28 2017

Formula

a(n) = a(n-1) +a(n-4) -a(n-6) +a(n-7) -3*a(n-9) +4*a(n-10) +5*a(n-12) -2*a(n-13) +3*a(n-15) -8*a(n-16) +a(n-18) -4*a(n-19) +3*a(n-21) -4*a(n-22) -3*a(n-24) -5*a(n-27) +8*a(n-28) +7*a(n-30) -2*a(n-31) -9*a(n-33) +2*a(n-34) +5*a(n-36) +4*a(n-37) +a(n-39) -6*a(n-40) -3*a(n-42) +2*a(n-43) +2*a(n-45) +a(n-46) -a(n-48) -a(n-49) +a(n-52).
G.f.: (1 -x^4 -x^5 -x^7 -x^8 +2*x^9 -x^10 -3*x^12 -x^13 -2*x^15 +3*x^16 +3*x^17 +2*x^18 -x^20 -4*x^21 +x^23 +3*x^24 +3*x^25 +x^27 -4*x^28 -x^29 -2*x^30 +x^31 +2*x^33 +x^34 -x^36 -x^37 +x^40 )/((1-x-x^4)*(1-x^9-x^12)*(1 +x^6 +4*x^9 -4*x^12 -2*x^15 +4*x^18 -3*x^21 -3*x^24 +7*x^27 -6*x^30 +3*x^33 -x^36)).
a(3*k) = (A003269(k))^3,
a(3*k+1) = (A003269(k))^2 * A003269(k+1),
a(3*k+2) = A003269(k) * (A003269(k+1))^2.

A224812 Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 63, 81, 108, 144, 192, 256, 336, 441, 567, 729, 918, 1156, 1462, 1849, 2365, 3025, 3905, 5041, 6532, 8464, 10948, 14161, 18207, 23409, 29988, 38416, 49196, 63001, 80822, 103684, 133308, 171396, 220662, 284089, 365638, 470596, 605052, 777924, 999306, 1283689, 1648515
Offset: 0

Views

Author

Vladimir Baltic, May 18 2013

Keywords

Comments

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=10, I={-2,0,10}.

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

Programs

  • Mathematica
    CoefficientList[Series[-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)
  • PARI
    x='x+O('x^50); Vec(-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1))) \\ G. C. Greubel, Oct 28 2017

Formula

a(n) = a(n-1) +a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11) +3*a(n-12) -2*a(n-13) +2*a(n-14) -a(n-15) +a(n-16) -2*a(n-19) +a(n-20) -2*a(n-21) -3*a(n-24) +a(n-25) -2*a(n-26) +a(n-31) +a(n-36).
G.f.: -(x+1) *(x^23 -x^22 +x^21 -x^20 +x^19 -x^13 +x^12 -3*x^11 +3*x^10 -3*x^9 +2*x^8 -2*x^7 +x^6 -x^5 +x^4 -x^3 +x^2 -x +1)/ ((x^6 +x -1) *(x^30 +x^24 -2*x^20 -2*x^18 -x^14 -2*x^12 +x^10 +x^8 +x^6+1) ).
a(2*k) = (A005708(k))^2, a(2*k+1) = A005708(k) * A005708(k+1).

A224813 Number of subsets of {1,2,...,n-12} without differences equal to 2, 4, 6, 8, 10 or 12.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 80, 100, 130, 169, 221, 289, 374, 484, 616, 784, 980, 1225, 1505, 1849, 2279, 2809, 3498, 4356, 5478, 6889, 8715, 11025, 13965, 17689, 22344, 28224, 35448, 44521, 55704, 69696, 87120, 108900, 136290, 170569, 213934, 268324, 337218, 423801, 533169, 670761, 843570
Offset: 0

Views

Author

Vladimir Baltic, May 18 2013

Keywords

Comments

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=12, I={-2,0,12}.

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

Programs

  • Mathematica
    CoefficientList[Series[-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 1)), {x, 0, 1000}], x] (* G. C. Greubel, Oct 28 2017 *)
  • PARI
    x='x+O('x^50); Vec(-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 1))) \\ G. C. Greubel, Oct 28 2017

Formula

a(n) = a(n-1) +a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11) -a(n-12) +a(n-13) +3*a(n-14) -2*a(n-15) +2*a(n-16) -a(n-17) +a(n-18) -3*a(n-21) +2*a(n-22) -4*a(n-23) +2*a(n-24) -3*a(n-25) -3*a(n-28) +a(n-29) -2*a(n-30) +3*a(n-35) -a(n-36) +3*a(n-37) +a(n-42) -a(n-49).
G.f.: -(-1 +x^7 +x^9 +x^11 +2*x^14 +x^16 -2*x^21 -2*x^23 -x^28 +x^35)/( (x^7+x-1) *(x^42 -x^36 -2*x^30 -3*x^28 +2*x^24 +2*x^22 +x^18 +2*x^16 +3*x^14 -x^12 -x^10 -x^8 -1) ).
a(2*k) = (A005709(k))^2, a(2*k+1) = A005709(k) * A005709(k+1).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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