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.

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A079974 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=4, I={0,2}.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 2, 5, 5, 9, 10, 16, 20, 30, 39, 56, 75, 106, 144, 201, 275, 382, 525, 727, 1001, 1384, 1908, 2636, 3636, 5021, 6928, 9565, 13200, 18222, 25149, 34715, 47914, 66137, 91285, 126001, 173914, 240052, 331336, 457338, 631251, 871304, 1202639
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Number of compositions (ordered partitions) of n into elements of the set {2,4,5}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

Formula

Recurrence: a(n) = a(n-2)+a(n-4)+a(n-5).
G.f.: -1/(x^5+x^4+x^2-1)

A079975 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=4, I={3}.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 49, 93, 175, 331, 625, 1180, 2229, 4209, 7949, 15012, 28350, 53540, 101111, 190950, 360613, 681024, 1286127, 2428875, 4586976, 8662591, 16359466, 30895160, 58346092, 110187694, 208091537, 392984789, 742159180
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Number of compositions (ordered partitions) of n into elements of the set {1,2,3,5}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Formula

Recurrence: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-5).
G.f.: -1/(x^5+x^3+x^2+x-1).

A079982 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-1,0,1,2}.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 4, 1, 1, 1, 1, 10, 10, 7, 7, 11, 20, 50, 40, 49, 61, 85, 175, 225, 265, 323, 461, 665, 1085, 1310, 1728, 2290, 3171, 4767, 6489, 8618, 11374, 15751, 21813, 31263, 41749, 56596, 76735, 105514, 147726, 202628, 276079, 374275
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0,1}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

  • Mathematica
    LinearRecurrence[{0,1,0,0,2,3,-1,-3,1,-1,-1,-3,1,3,-2,0,0,1,0,-1},{1,0,0,0,0,1,1,0,0,0,1,4,1,1,1,1,10,10,7,7},60] (* Harvey P. Dale, Dec 19 2021 *)

Formula

Recurrence: a(n) = a(n-2)+2*a(n-5)+3*a(n-6)-a(n-7)-3*a(n-8)+a(n-9)-a(n-10)-a(n-11)-3*a(n-12)+a(n-13)+3*a(n-14)-2*a(n-15)+a(n-18)-a(n-20).
G.f.: -(x^14-x^12+x^9-2*x^8+2*x^6+x^5+x^2-1)/(x^20-x^18+2*x^15-3*x^14-x^13+3*x^12+x^11+x^10-x^9+3*x^8+x^7-3*x^6-2*x^5-x^2+1).

A079983 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,1,2}.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 12, 20, 35, 60, 114, 207, 375, 671, 1213, 2180, 3954, 7139, 12892, 23250, 41996, 75793, 136891, 247133, 446211, 805505, 1454390, 2625744, 4740788, 8559108, 15453182, 27899503, 50371415, 90942627, 164192549, 296440115
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,2}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Formula

a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-5) +4*a(n-6) -4*a(n-7) -3*a(n-8) -3*a(n-9) +3*a(n-10) +3*a(n-11) -5*a(n-12) +4*a(n-13) +4*a(n-14) +a(n-15) -2*a(n-16) -a(n-17) +a(n-18) -a(n-19) -a(n-20)
G.f.: -(x^2-1)*(x^12+2*x^9-x^6-2*x^3+1)/(x^20 +x^19 -x^18 +x^17 +2*x^16 -x^15 -4*x^14 -4*x^13 +5*x^12 -3*x^11 -3*x^10 +3*x^9 +3*x^8 +4*x^7 -4*x^6 +x^5 -x^3 -x^2 -x+1)

A079984 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-1,1,2}.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 16, 26, 43, 80, 148, 264, 465, 816, 1444, 2588, 4619, 8214, 14591, 25903, 46071, 82015, 145904, 259492, 461408, 820468, 1459332, 2595687, 4616613, 8210719, 14602409, 25970414, 46189613, 82149988, 146106304, 259853016
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,1}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

  • Mathematica
    CoefficientList[Series[-(x^14-x^12+2x^11-x^9-x^8+x^6-x^5+2x^3+x^2-1)/ ((x^19-x^17+2x^16x^15+2x^14-5x^13+x^12+3x^11-5x^10+x^9+x^8+ 3x^7+x^6- 4x^5+ 3x^4- 2x^3+ x^2-2x+1)(x+1)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,1,1,-1,1,3,-4,-4,-2,4,2,-4,4,3,-1,-1,-1,1,-1,-1},{1,1,1,1,2,5,10,16,26,43,80,148,264,465,816,1444,2588,4619,8214,14591},50] (* Harvey P. Dale, Feb 07 2015 *)

Formula

a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-4) +a(n-5) +3*a(n-6) -4*a(n-7) -4*a(n-8) -2*a(n-9) +4*a(n-10) +2*a(n-11) -4*a(n-12) +4*a(n-13) +3*a(n-14) -a(n-15) -a(n-16) -a(n-17) +a(n-18) -a(n-19) -a(n-20)
G.f.: -(x^14-x^12+2*x^11-x^9-x^8+x^6-x^5+2*x^3+x^2-1)/((x^19 -x^17 +2*x^16 -x^15 +2*x^14 -5*x^13 +x^12 +3*x^11 -5*x^10 +x^9 +x^8 +3*x^7 +x^6 -4*x^5 +3*x^4 -2*x^3 +x^2 -2*x+1) *(x+1))

A079985 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-1,0,1}.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 4, 0, 4, 10, 29, 40, 64, 88, 221, 420, 800, 1280, 2336, 4260, 8325, 15000, 27200, 48360, 89541, 164430, 303300, 549120, 1001156, 1824342, 3350169, 6122640, 11189504, 20391280, 37266329, 68097480, 124548224, 227452928, 415474816
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Formula

a(n) = a(n-1) +a(n-2) -a(n-3) +3*a(n-4) +2*a(n-5) +2*a(n-6) -6*a(n-7) -2*a(n-8) +2*a(n-9) -3*a(n-10) -a(n-11) -a(n-12) +a(n-13) +a(n-14)
G.f.: -(x^8+x^7-x^6-x^5-2*x^4+x^3-x^2-x+1)/(x^14 +x^13 -x^12 -x^11 -3*x^10 +2*x^9 -2*x^8 -6*x^7 +2*x^6 +2*x^5 +3*x^4 -x^3 +x^2 +x-1)

A079986 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,0,2}.

Original entry on oeis.org

1, 0, 1, 0, 4, 0, 16, 0, 49, 0, 169, 0, 576, 0, 1936, 0, 6561, 0, 22201, 0, 75076, 0, 254016, 0, 859329, 0, 2907025, 0, 9834496, 0, 33269824, 0, 112550881, 0, 380757169, 0, 1288092100, 0, 4357584144, 0, 14741602225
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

a(n)=( A000073(k+2) )^2 if n=2k, a(n)=0 otherwise.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

  • Mathematica
    LinearRecurrence[{0,2,0,3,0,6,0,-1,0,0,0,-1},{1,0,1,0,4,0,16,0,49,0,169,0},50] (* Harvey P. Dale, Nov 03 2015 *)

Formula

a(n) = 2*a(n-2)+3*a(n-4)+6*a(n-6)-a(n-8)-a(n-12).
G.f.: -(x^6+x^4+x^2-1)/(x^12+x^8-6*x^6-3*x^4-2*x^2+1)

A079987 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-1,0,2}.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 5, 7, 15, 29, 49, 84, 149, 268, 484, 855, 1508, 2684, 4784, 8516, 15134, 26873, 47782, 85004, 151149, 268704, 477685, 849299, 1510163, 2685089, 4773851, 8487625, 15090786, 26831239, 47705352, 84818268, 150803857, 268125092
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,0,1}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Programs

  • Mathematica
    LinearRecurrence[{2,-2,3,-2,4,-1,-2,4,-5,4,-6,3,0,-2,1,-2,2,-1},{1,0,0,1,2,3,5,7, 15,29, 49,84,149,268,484,855,1508,2684},40] (* Harvey P. Dale, Apr 29 2011 *)

Formula

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

A079988 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={0,1,2}.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 3, 3, 6, 10, 29, 39, 61, 101, 179, 335, 566, 928, 1575, 2705, 4747, 8117, 13782, 23464, 40216, 69209, 118650, 202712, 346508, 593180, 1016874, 1741871, 2981190, 5101520, 8733466, 14956519, 25611753, 43847283, 75061015, 128505176
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Formula

a(n) = a(n-2) +a(n-3) +a(n-4) +4*a(n-5) +4*a(n-6) -a(n-7) -3*a(n-8) -2*a(n-9) +a(n-10) -4*a(n-11) -3*a(n-12) +a(n-13) +4*a(n-14) -a(n-16) +a(n-17) +a(n-18) -a(n-20) .
G.f.: -(x^14-x^12 -x^11+x^9 -2*x^8+2*x^6 +x^5+x^3 +x^2-1)/( x^20 -x^18 -x^17 +x^16 -4*x^14 -x^13 +3*x^12 +4*x^11 -x^10 +2*x^9 +3*x^8 +x^7 -4*x^6 -4*x^5 -x^4 -x^3 -x^2+1).

A079990 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-1,2}.

Original entry on oeis.org

1, 1, 1, 2, 6, 16, 36, 73, 157, 353, 797, 1776, 3916, 8636, 19145, 42504, 94286, 208948, 462907, 1025863, 2274069, 5040891, 11173063, 24763854, 54886846, 121655063, 269646786, 597664017, 1324697483, 2936135519, 6507831521, 14424377636
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Also, number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,1}.

References

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

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Formula

a(n) = a(n-1) +a(n-2) +2*a(n-3) +a(n-4) +4*a(n-5) +7*a(n-6) -4*a(n-7) -a(n-8) +a(n-9) +7*a(n-10) +3*a(n-11) -7*a(n-12) +2*a(n-13) +3*a(n-14) -3*a(n-16) -2*a(n-17) +a(n-18) -a(n-19) -a(n-20).
G.f.: -(x^14-x^12+2*x^11-x^9+2*x^6-x^5+2*x^3+x^2-1)/(x^20 +x^19 -x^18 +2*x^17 +3*x^16 -3*x^14 -2*x^13 +7*x^12 -3*x^11 -7*x^10 -x^9 +x^8 +4*x^7 -7*x^6 -4*x^5 -x^4 -2*x^3 -x^2 -x+1)
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