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 1-10 of 10 results.

A017877 Expansion of 1/(1 - x^9 - x^10).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 1, 8, 28, 56
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9 and 10. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Column k=9 of A306713.
Cf. A017876, A017878, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

Programs

  • Magma
    m:=80; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 10]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
  • SageMath
    def A017877_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^9-x^(10)) ).list()
A017877_list(85) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) + a(n-10) for n > 9. - Vincenzo Librandi, Jul 01 2013
a(n) = Sum_{k=0..floor(n/9)} binomial(k,n-9*k). - Seiichi Manyama, Oct 01 2024

A017886 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 19, 23, 28, 34, 41, 49, 59, 72, 86, 102, 122, 146, 175, 210, 252, 303, 366, 441, 529, 635, 762, 914, 1096, 1314, 1576, 1893, 2275
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions of n into parts 9, 10, 11, ..., 19. - Joerg Arndt, Oct 12 2014

Links

Crossrefs

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017887.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 19]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)
  • SageMath
    def A017886_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(20)) ).list()
A017886_list(70) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) +a(n-17) +a(n-18) +a(n-19) for n>18. - Vincenzo Librandi, Jul 01 2013
a(n) = a(n-1) +a(n-9) -a(n-20) for n>19. - Tani Akinari, Sep 29 2014

A017878 Expansion of 1/(1-x^9-x^10-x^11).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0, 1, 3, 6, 7, 6, 3, 1, 0, 0, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 6, 7, 21, 50, 90, 126, 141, 126, 90, 51, 28, 34, 78, 161
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10 and 11. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 11]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1},{1,0,0,0,0,0,0,0,0,1,1},70] (* Harvey P. Dale, May 25 2023 *)
  • SageMath
    def A017878_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(12)) ).list()
A017878_list(80) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) for n>10. - Vincenzo Librandi, Jul 01 2013

A017879 Expansion of 1/(1-x^9-x^10-x^11-x^12).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 3, 6, 10, 12, 12, 10, 6, 3, 2, 4, 10, 20, 31, 40, 44, 40, 31, 21, 15, 19, 36, 65, 101, 135, 155, 155, 136, 107, 86, 91, 135, 221, 337, 456, 546
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11 and 12. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1/(1-x^9 -x^10 -x^11 -x^12), {x,0,70}], x] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1}, {1,0,0,0,0,0,0,0,0,1,1,1}, 70] (* Harvey P. Dale, Apr 29 2013 *)
CoefficientList[Series[1/(1 - Total[x^Range[9, 12]]), {x,0,70}], x] (* Vincenzo Librandi, Jul 01 2013 *)
  • SageMath
    def A017879_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(13)) ).list()
A017879_list(85) # G. C. Greubel, Sep 25 2024

Formula

a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=1, a(10)=1, a(11)=1; for n>11, a(n) = a(n-9)+a(n-10)+a(n-11)+a(n-12). - Harvey P. Dale, Apr 29 2013

A017880 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 11, 10, 13, 21, 35, 52, 68, 80, 85, 81, 73, 67, 70, 90, 131, 189, 256, 320, 366, 387, 386, 376, 381, 431, 547
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12 and 13. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017879, A017881, A017882, A017883, A017884, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 13]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,1,1,1,1},70] (* Harvey P. Dale, Apr 03 2018 *)
  • SageMath
    def A017880_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(14)) ).list()
A017880_list(80) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) for n>12. - Vincenzo Librandi, Jul 01 2013

A017881 Expansion of 1/(1 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 3, 4, 6, 10, 15, 21, 25, 27, 27, 26, 25, 25, 30, 41, 59, 81, 104, 125, 141, 151, 155, 160, 174, 206, 261, 340, 440, 551, 661, 757, 836, 906, 987
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13 and 14. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017879, A017880, A017882, A017883, A017884, A017885, A017886.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (1-x)/(1-x-x^9+x^(15)) )); // G. C. Greubel, Sep 25 2024
  • Mathematica
    CoefficientList[Series[1/(1-Total[x^Range[9,14]]),{x,0,60}],x] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,1,1,1,1,1}, 60] (* Harvey P. Dale, Feb 27 2012 *)
  • SageMath
    def A017881_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(15)) ).list()
A017881_list(80) # G. C. Greubel, Sep 25 2024

Formula

a(0)=1, a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=a(7)=a(8)=0, a(9)=a(10)=a(11)=a(12)= a(13)=1, a(n) = a(n-9) + a(n-10) + a(n-11) + a(n-12) + a(n-13) + a(n-14). - Harvey P. Dale, Feb 27 2012

A017882 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 5, 6, 8, 11, 15, 21, 28, 33, 36, 38, 40, 43, 48, 56, 71, 94, 122, 152, 182, 211, 239, 266, 294, 332, 390, 474, 586, 725, 888, 1071, 1266, 1466
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14 and 15. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017879, A017880, A017881, A017883, A017884, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15)));  // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 15]]),{x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
  • SageMath
    def A017882_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(16)) ).list()
A017882_list(80) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) for n>14. - Vincenzo Librandi, Jul 01 2013

A017883 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 7, 8, 10, 13, 17, 22, 28, 36, 42, 47, 52, 58, 66, 77, 92, 112, 141, 176, 215, 257, 302, 351, 406, 470, 546, 645, 774, 937, 1136, 1372, 1646
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14, 15 and 16. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017884, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 16]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)
  • SageMath
    def A017883_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(17)) ).list()
A017883_list(65) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) for n>15. - Vincenzo Librandi, Jul 01 2013

A017884 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 12, 15, 19, 24, 30, 37, 45, 53, 61, 70, 81, 95, 113, 136, 165, 201, 245, 296, 354, 420, 496, 585, 691, 819, 975, 1167, 1402, 1686, 2025
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14, 15, 16 and 17. - Ilya Gutkovskiy, May 27 2017

Links

Crossrefs

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017883, A017885, A017886.

Programs

  • Magma
    m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17))); // Vincenzo Librandi, Jul 01 2013
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[9, 17]]), {x, 0, 60}], x] (* Harvey P. Dale, Sep 12 2012 *)
  • SageMath
    def A017884_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(18)) ).list()
A017884_list(70) # G. C. Greubel, Sep 25 2024

Formula

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) +a(n-17) for n>16. - Vincenzo Librandi, Jul 01 2013

A274165 Number of real integers in n-th generation of tree T(i/3) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 17, 21, 26, 32, 39, 47, 57, 67, 79, 93, 110, 131, 157, 189, 228, 276, 332, 399, 478, 571, 681, 812, 969, 1158, 1387, 1662, 1994, 2393, 2871, 3442, 4123, 4935, 5904, 7063, 8449, 10111
Offset: 0

Views

Author

Clark Kimberling, Jun 12 2016

Keywords

Comments

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
See A274142 for a guide to related sequences.
a(n) = A017885(n+7) for 2 <= n < 85, but a(85) = 1314173 differs from A017885(92) = 1314172. - Georg Fischer, Oct 30 2018

Examples

			If r = i/3, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 1.

Links

Crossrefs

Cf. A274142.

Programs

  • Mathematica
    z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> I/3, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

Extensions

More terms from Kenny Lau, Jun 30 2017
Showing 1-10 of 10 results.

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

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