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 1-5 of 5 results.

A055991 a(n) is its own 4th difference.

Original entry on oeis.org

1, 5, 19, 69, 250, 907, 3292, 11949, 43371, 157422, 571388, 2073943, 7527704, 27322992, 99173120, 359964521, 1306548149, 4742323107, 17213011605, 62477347458, 226771411939, 823102698260, 2987581397893, 10843899100203
Offset: 1

Views

Author

Henry Bottomley, Jun 02 2000

Keywords

Comments

a(n) is the number of distinct matrix products in (A+B+C+D+E)^n where A,B,C and D all commute with each other, but not with E. - Paul D. Hanna and Max Alekseyev, Feb 01 2006
Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry, Feb 02 2006
Quadrisection of A003269: a(n)=A003269(4n-1). - Paul Barry, Feb 02 2006
From Gary W. Adamson, Apr 23 2009: (Start)
Equals the INVERT transform of the tetrahedral series.
a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End)

Links

Crossrefs

Cf. A055988, A055989, A055990 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Programs

  • Magma
    I:=[1, 5, 19, 69]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012
  • Mathematica
    LinearRecurrence[{5,-6,4,-1},{1,5,19,69},30] (* Harvey P. Dale, Feb 27 2013 *)

Formula

a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055990(n) = A055988(n+1)-A055988(n) = A055989(n+1)-2*A055989(n)+A055989(n-1).
Letting a(0)=1, we have a(n)=sum(u=0, n-1, sum(v=0, u, sum(w=0, v, sum(x=0, w, a(x))))) for n>0. - Benoit Cloitre, Jan 26 2003
a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic, Mar 23 2003
a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - Paul Barry, Feb 02 2006
G.f.: x/(1-5x+6x^2-4x^3+x^4). - Paul Barry, Feb 02 2006

A055988 Sequence is its own 4th difference.

Original entry on oeis.org

1, 2, 7, 26, 95, 345, 1252, 4544, 16493, 59864, 217286, 788674, 2862617, 10390321, 37713313, 136886433, 496850954, 1803399103, 6545722210, 23758733815, 86236081273, 313007493212, 1136110191472, 4123691589365, 14967590689568
Offset: 1

Views

Author

Henry Bottomley, Jun 02 2000

Keywords

Comments

Row sums of Riordan array (1/(1-x), x/(1-x)^4), A109960. - Paul Barry, Jul 06 2005

Links

Crossrefs

Cf. A055989, A055990, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Programs

  • Magma
    I:=[1, 2, 7, 26]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012
  • Mathematica
    CoefficientList[Series[(1-x)^3/(1-5x+6x^2-4x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Apr 05 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,2,7,26},30] (* Harvey P. Dale, Jan 15 2017 *)

Formula

a(n) = 5a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4) = a(n-1) + A055991(n-1) = A055989(n) - A055989(n-1) = A055990(n) - 2*A055990(n-1) + A055990(n-2).
From Paul Barry, Jul 06 2005: (Start)
G.f.: (1-x)^3/(1 - 5x + 6x^2 - 4x^3 + x^4);
a(n) = Sum_{k=0..n} binomial(n+3k, 4k). (End)

Extensions

More terms from James Sellers, Jun 05 2000

A369837 Number of compositions of 5*n-2 into parts 1 and 5.

Original entry on oeis.org

1, 5, 20, 80, 325, 1326, 5411, 22076, 90061, 367411, 1498887, 6114853, 24946129, 101770120, 415180936, 1693770328, 6909898016, 28189589705, 115002126790, 469162173146, 1913991948274, 7808313175575, 31854760257925, 129954540535600, 530161974821876
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A079675, A369836, A369838, A369839.
Cf. A003520, A055990.

Programs

  • Mathematica
    LinearRecurrence[{6, -10, 10, -5, 1}, {1, 5, 20, 80, 325}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2+4*k, n-1-k));

Formula

a(n) = A003520(5*n-2).
a(n) = Sum_{k=0..n} binomial(n+2+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)/((1-x)^5 - x).

A055989 a(n) is its own 4th difference.

Original entry on oeis.org

1, 3, 10, 36, 131, 476, 1728, 6272, 22765, 82629, 299915, 1088589, 3951206, 14341527, 52054840, 188941273, 685792227, 2489191330, 9034913540, 32793647355, 119029728628, 432037221840, 1568147413312, 5691839002677, 20659429692245, 74986666876571, 272175964826781
Offset: 1

Views

Author

Henry Bottomley, Jun 02 2000

Keywords

Comments

Number of compositions of 4*n-3 into parts 1 and 4. - Seiichi Manyama, Feb 03 2024

Links

Crossrefs

Cf. A055988, A055990, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.
Cf. A003269.

Programs

  • Magma
    I:=[1, 3, 10, 36]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012
  • Mathematica
    CoefficientList[Series[(1-x)^2/(1-5*x+6*x^2-4*x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Apr 05 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,3,10,36},30] (* Harvey P. Dale, Jan 10 2014 *)

Formula

a(n) = 5*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) = a(n-1) + A055988(n) = A055990(n) - A055990(n-1) = A055991(n) - 2*A055991(n-1) + A055991(n-2).
G.f.: x*(1-x)^2/(1 - 5*x + 6*x^2 - 4*x^3 + x^4). - Colin Barker Apr 04 2012

Extensions

More terms from James Sellers, Jun 05 2000

A338995 Triangle T(n,m):=binomial(n+3*m+2,n-m).

Original entry on oeis.org

1, 3, 1, 6, 7, 1, 10, 28, 11, 1, 15, 84, 66, 15, 1, 21, 210, 286, 120, 19, 1, 28, 462, 1001, 680, 190, 23, 1, 36, 924, 3003, 3060, 1330, 276, 27, 1, 45, 1716, 8008, 11628, 7315, 2300, 378, 31, 1, 55, 3003, 19448, 38760, 33649, 14950, 3654, 496, 35, 1, 66, 5005, 43758, 116280, 134596, 80730, 27405, 5456, 630, 39, 1
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 17 2020

Keywords

Comments

Riordan Array (1/(1-x)^3,x/(1-x)).
Sum_{m=0..n} T(n,m) = A055990(n-1).

Examples

			1,
3, 1,
6, 7, 1,
10, 28, 11, 1,
15, 84, 66, 15, 1,
21, 210, 286, 120, 19, 1,
28, 462, 1001, 680, 190, 23, 1

Crossrefs

Cf. A000217 (1st column), A007318, A055990 (row sums).

Formula

G.f.: 1/(1-x)^3/(1-y*x/(1-x)^4).
Showing 1-5 of 5 results.

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