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 1-7 of 7 results.

A145112 Numbers of length n binary words with fewer than 4 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1728, 3334, 6430, 12398, 23902, 46077, 88821, 171213, 330029, 636157, 1226238, 2363656, 4556100, 8782172, 16928188, 32630139, 62896623, 121237147, 233692123, 450456059, 868281980, 1673667338, 3226097530, 6218502938
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(6) = 63 = 2^6-1, because 100001 is the only binary word of length 6 with not less than 4 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

4th column of A145111.
Cf. A242234.

Programs

  • Maple
    a:= n-> (Matrix([[2, 1$5]]). Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$2, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
  • Mathematica
    CoefficientList[Series[(1 - x + x^5) / (1 - 3 x + 2 x^2 + x^5 - x^6), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Formula

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

A145113 Numbers of length n binary words with fewer than 5 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7424, 14598, 28702, 56430, 110942, 218110, 428797, 842997, 1657293, 3258157, 6405373, 12592637, 24756478, 48669960, 95682628, 188107100, 369808828, 727025020, 1429293563, 2809917167, 5524151707
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(7) = 127 = 2^7-1, because 1000001 is the only binary word of length 7 with not less than 5 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

5th column of A145111.

Programs

  • Maple
    a:= n-> (Matrix([[2, 1$6]]). Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$3, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
  • Mathematica
    CoefficientList[Series[(1 - x + x^6) / (1 - 3 x + 2 x^2 + x^6 - x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Formula

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

A145114 Numbers of length n binary words with fewer than 6 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30976, 61446, 121886, 241774, 479582, 951294, 1886974, 3742973, 7424501, 14727117, 29212461, 57945341, 114939389, 227991805, 452240638, 897056776, 1779386436, 3529560412, 7001175484
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(8) = 255 = 2^8-1, because 10000001 is the only binary word of length 8 with not less than 6 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

6th column of A145111.

Programs

  • Maple
    a:= n-> (Matrix([[2, 1$7]]). Matrix(8, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$4, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);
  • Mathematica
    CoefficientList[Series[(1 - x + x^7) / (1 - 3 x + 2 x^2 + x^7 - x^8), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{3,-2,0,0,0,0,-1,1},{1,2,4,8,16,32,64,128},40] (* Harvey P. Dale, Mar 13 2023 *)

Formula

G.f.: (1-x+x^7)/(1-3*x+2*x^2+x^7-x^8).

A145115 Numbers of length n binary words with fewer than 7 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63743, 126976, 252934, 503838, 1003630, 1999198, 3982334, 7932670, 15801598, 31476221, 62699509, 124895181, 248786733, 495574269, 987166205, 1966399741, 3916997885, 7802519550
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(9) = 511 = 2^9-1, because 100000001 is the only binary word of length 9 with not less than 7 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

7th column of A145111.

Programs

  • Maple
    a:= n-> (Matrix([[2, 1$8]]). Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$5, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);
  • Mathematica
    CoefficientList[Series[(1 - x + x^8) / (1 - 3 x + 2 x^2 + x^8 - x^9), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Formula

G.f.: (1-x+x^8)/(1-3*x+2*x^2+x^8-x^9).

A145116 Numbers of length n binary words with fewer than 8 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2043, 4079, 8143, 16255, 32447, 64767, 129279, 258047, 515072, 1028102, 2052126, 4096110, 8175966, 16319486, 32574206, 65019134, 129780222, 259045373, 517062645, 1032073165, 2060050221, 4111924477, 8207529469
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(10) = 1023 = 2^10-1, because 1000000001 is the only binary word of length 10 with not less than 8 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

8th column of A145111.

Programs

  • Maple
    a:= n-> (Matrix([[2, 1$9]]). Matrix(10, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$6, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);
  • Mathematica
    CoefficientList[Series[(1 - x + x^9) / (1 - 3 x + 2 x^2 + x^9 - x^10), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Formula

G.f.: (1-x+x^9)/(1-3*x+2*x^2+x^9-x^10).

A145117 Numbers of length n binary words with fewer than 9 0-digits between any pair of consecutive 1-digits.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4091, 8175, 16335, 32639, 65215, 130303, 260351, 520191, 1039359, 2076672, 4149254, 8290334, 16564334, 33096030, 66126846, 132123390, 263986430, 527452670, 1053865982, 2105655293, 4207161333, 8406032333
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2008

Keywords

Examples

			a(11) = 2047 = 2^11-1, because 10000000001 is the only binary word of length 11 with not less than 9 0-digits between any pair of consecutive 1-digits.

Links

Crossrefs

9th column of A145111.

Programs

  • Maple
    a:= n-> (Matrix([[2,1$10]]). Matrix(11, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$7, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);
  • Mathematica
    CoefficientList[Series[(1 - x + x^10) / (1 - 3 x + 2 x^2 + x^10 - x^11), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{3,-2,0,0,0,0,0,0,0,-1,1},{1,2,4,8,16,32,64,128,256,512,1024},40] (* Harvey P. Dale, Sep 24 2016 *)

Formula

G.f.: (1-x+x^10)/(1-3*x+2*x^2+x^10-x^11).

A216274 Square array A(n,k) = maximal number of regions into which k-space can be divided by n hyperplanes (k >= 1, n >= 0), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 7, 5, 1, 2, 4, 8, 11, 6, 1, 2, 4, 8, 15, 16, 7, 1, 2, 4, 8, 16, 26, 22, 8, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 2, 4, 8, 16, 32, 64, 120, 163, 130, 56, 12
Offset: 0

Views

Author

Frank M Jackson, Mar 16 2013

Keywords

Comments

For all fixed k, the sequences A(n,k) are "complete" (sic).
This array is similar to A145111 with first variation at 34th term.

Examples

			Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2, ...
  3,  4,  4,  4,  4,  4, ...
  4,  7,  8,  8,  8,  8, ...
  5, 11, 15, 16, 16, 16, ...
  6, 16, 26, 31, 32, 32, ...
So the maximal number of pieces into which a cube can be divided after 5 planar cuts is A(5,3) = 26.

Links

  • Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Crossrefs

Cf. A000124, A000125, A059214.

Programs

  • Mathematica
    getvalue[n_, k_] := Sum[Binomial[n, i], {i, 0, k}]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@IntegerPartitions[#1+dim-1, {dim}], 1] &, maxHeight], 1]; pairs = lexicographicLattice[{2, 12}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}]

Formula

A(k,n) = Sum_{i=0..k} C(n, i), k >=1, n >= 0.

Extensions

Edited by N. J. A. Sloane, May 20 2023
Showing 1-7 of 7 results.

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

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