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.

A168583 The number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly three nonempty parts.

Original entry on oeis.org

1, 4, 16, 58, 196, 634, 1996, 6178, 18916, 57514, 174076, 525298, 1582036, 4758394, 14299756, 42948418, 128943556, 387027274, 1161475036, 3485211538, 10457207476, 31374768154, 94130595916, 282404370658, 847238277796, 2541765165034, 7625396158396
Offset: 3

Views

Author

Martin Griffiths, Nov 30 2009

Keywords

Comments

The number of ways of partitioning the multiset {1, 1, 2, 3, ..., n-1} into exactly two, four and five nonempty parts are given in A083329, A168584 and A168585, respectively.

Examples

			The partitions of {1,1,2,3} into exactly three nonempty parts are {{1},{1},{2,3}}, {{1},{2},{1,3}}, {{1},{3},{1,2}} and {{2},{3},{1,1}}.

Links

Crossrefs

Cf. A083329, A126644, A168584, A168585.

Programs

  • Magma
    [3^(n-2) - 3*2^(n-3) + 1: n in [3..35]]; // Vincenzo Librandi, Dec 12 2015
  • Maple
    A168583:=n->3^(n-2)-3*2^(n-3)+1: seq(A168583(n), n=3..40); # Wesley Ivan Hurt, Dec 12 2015
  • Mathematica
    f1[n_] := 3^(n - 2) - 3 2^(n - 3) + 1; Table[f1[n], {n, 3, 25}]

Formula

For a>=3, a(n) = 3^(n-2) - 3*2^(n-3) + 1.
E.g.f.: 3*e^(3x) - 3*e^(2x) + e^x (shifted).
O.g.f.: x^3*(1-2x+3x^2)/((1-x)*(1-2x)*(1-3x)).
a(n) = A126644(n-3). - R. J. Mathar, Dec 11 2009

A126718 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3, at least one of digits 4,5, at least one of digits 6,7 and at least one of digits 8,9.

Original entry on oeis.org

7, 43, 235, 1171, 5467, 24403, 105595, 447091, 1864027, 7686163, 31440955, 127865011, 517788187, 2090186323, 8417944315, 33843570931, 135890057947, 545108340883, 2185079263675, 8754257900851, 35058860433307, 140360940805843, 561820285607035
Offset: 1

Views

Author

Aleksandar M. Janjic and Milan Janjic, Feb 13 2007

Keywords

Links

Crossrefs

Cf. A125910, A125945, A125946, A125947, A125948, A125880, A125630, A125987, A125904, A125858, A125909, A125908, A126646, A126645, A126644, A126643, A126642, A126641, A126640, A126639, A126635, A126634, A126633, A126632, A126631, A126628, A126627.

Programs

  • Magma
    [8*4^n-12*3^n+6*2^n-1: n in [1..30]]; // Vincenzo Librandi, May 31 2011
  • Maple
    a:=n->8*4^n-12*3^n+6*2^n-1;
  • Mathematica
    LinearRecurrence[{10,-35,50,-24},{7, 43, 235, 1171},23] (* James C. McMahon, Dec 27 2024 *)
  • PARI
    Vec(-x*(24*x^3-50*x^2+27*x-7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

Formula

a(n) = 8*4^n - 12*3^n + 6*2^n - 1.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4). - Colin Barker, Feb 22 2015
G.f.: -x*(24*x^3 - 50*x^2 + 27*x - 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Feb 22 2015

A140420 Binomial transform of 0, 1, 1, 7, 7, 31, 31, ..., zero followed by duplicated A083420.

Original entry on oeis.org

0, 1, 3, 13, 45, 151, 483, 1513, 4665, 14251, 43263, 130813, 394485, 1187551, 3570843, 10728913, 32219505, 96724051, 290303223, 871171813, 2614039725, 7843167751, 23531600403, 70598995513, 211805375145, 635432902651
Offset: 0

Views

Author

Paul Curtz, Jun 18 2008, corrected Jun 23 2008

Keywords

Links

Formula

a(n+1)-3a(n) = A099430(n).
O.g.f.: (3x^2-2x+1)x/((2x-1)(1+x)(3x-1)(1-x)). - R. J. Mathar, Jul 10 2008
a(n)+a(n+1)=A126644(n). - R. J. Mathar, Jul 10 2008

Extensions

More terms from R. J. Mathar, Jul 10 2008

A258800 The number of zeroless decimal numbers whose digital sum is n.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960, 4190597000, 8372936304, 16729373488, 33425781248
Offset: 0

Views

Author

Robert G. Wilson v, Jun 10 2015

Keywords

Comments

If you were to include decimal numbers that contain any number of zeros, then a(n) would be infinity. If on the other hand, you limit the number of zeros to some number, then a(n) is finite.

Examples

			a(0) = 0 since there exists no decimal number lacking a zero whose digital sum is zero.
a(1) = 1 since there exists only one zeroless decimal number whose digital sum is one and that number is 1.
a(2) = 2 since there exist only two zeroless decimal numbers whose digital sum is two and they are 2 & 11.
a(3) = 4 since there exist only four zeroless decimal numbers whose digital sum is three and they are 3, 21, 12 & 111.
a(4) = 8 since there exist only eight zeroless decimal numbers whose digital sum is four and they are 4, 31, 13, 22, 211, 121, 112 & 1111.

Crossrefs

Cf. A104144.
Cf. A125630, A125858, A125858, A125880, A125897, A125904, A125908, A125909, A125910, A125945, A125946, A125947, A125948, A126627, A126628, A126629, A126631, A126632, A126633, A126634, A126635, A126639, A126640, A126641, A126642, A126643, A126644, A126645, A126646, A126718.
Cf. A211072.

Programs

  • Mathematica
    CoefficientList[ Series[-1 + 1/(1 - x (1 + x + x^2) (1 + x^3 + x^6)), {x, 0, 36}], x]

Formula

a(n) = A104144(n+8) for n>0.
G.f.: -(x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9) = -1 + 1/(1-x(1 + x + x^2)(1 + x^3 + x^6)).
Showing 1-4 of 4 results.

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