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.

A373908 Number of compositions of 7*n into parts 2 and 7.

Original entry on oeis.org

1, 1, 2, 9, 38, 136, 452, 1495, 5031, 17114, 58282, 198032, 671856, 2278870, 7731892, 26238839, 89047335, 302191369, 1025487338, 3479970844, 11809261583, 40074827170, 135994407483, 461498426696, 1566098800484, 5314568565096, 18035031128780, 61202027710656
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373909, A373910, A373911, A373912.
Cf. A109961, A369840, A373904.
Cf. A369813.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+5*k, n-2*k));

Formula

a(n) = A369813(7*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,n-2*k).
a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^2/(1 - x)^6).

A109960 Number triangle binomial(n+3k,4k).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 9, 1, 1, 35, 45, 13, 1, 1, 70, 165, 91, 17, 1, 1, 126, 495, 455, 153, 21, 1, 1, 210, 1287, 1820, 969, 231, 25, 1, 1, 330, 3003, 6188, 4845, 1771, 325, 29, 1, 1, 495, 6435, 18564, 20349, 10626, 2925, 435, 33, 1, 1, 715, 12870, 50388, 74613, 53130
Offset: 0

Views

Author

Paul Barry, Jul 06 2005

Keywords

Comments

Riordan array (1/(1-x), x/(1-x)^4). Rows sums are A055988. Diagonal sums are A109961. Inverse array is A109962.

Examples

			Rows begin
1;
1,1;
1,5,1;
1,15,9,1;
1,35,45,13,1;
1,70,165,91,17,1;

Formula

Number triangle T(n, k)=binomial(n+3k, 4k)

A113067 Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); invert transform gives signed version of tetrahedral numbers A000292.

Original entry on oeis.org

0, -1, 4, -11, 28, -72, 188, -493, 1292, -3383, 8856, -23184, 60696, -158905, 416020, -1089155, 2851444, -7465176, 19544084, -51167077, 133957148, -350704367, 918155952, -2403763488, 6293134512, -16475640049, 43133785636, -112925716859, 295643364940, -774004377960
Offset: 0

Views

Author

Creighton Dement, Oct 13 2005

Keywords

Comments

Invert(a(n)) gives (0, -1, 4, -10, 20, -35, ...) = A000292 (with alternating signs).
Binomial(a(n)) gives (0, -1, 2, -2, 4, -7, 10, ...) = A094686 (with alternating signs, from 2nd term).
Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'

References

  • Creighton Dement, Floretion Integer Sequences (work in progress).

Links

Crossrefs

Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890.

Programs

  • Mathematica
    -x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* Jean-François Alcover, Jun 15 2017 *)
  • PARI
    concat(0, Vec(-x / ((1 + x + x^2)*(1 + 3*x + x^2)) + O(x^30))) \\ Colin Barker, May 11 2019
  • SageMath
    [((lucas_number1(n,3,1)-lucas_number1(n,1,1)))/(-2) for n in range(1,32)] # Zerinvary Lajos, Jul 06 2008

Formula

a(n) + a(n+1) + a(n+2) = (-1)^n *A001906(n+2) = (-1)^n*F(2n+4).
a(n) + 3*a(n+1) + 3*a(n+2) + a(n+3) = ((-1)^(n+1))*A109961(n+2).
(|a(n)|) = A290890(n) for n >= 0, this being the p-INVERT of (1,2,3,4,...), where p(S) = 1 - S^2. - Clark Kimberling, Aug 21 2017
a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, May 11 2019
2*a(n) = (-1)^n*A001906(n+1) - A049347(n). - R. J. Mathar, Sep 20 2020

A373904 a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k,n-2*k).

Original entry on oeis.org

1, 1, 2, 8, 30, 98, 303, 937, 2936, 9260, 29209, 91999, 289547, 911255, 2868341, 9029425, 28424456, 89478064, 281667368, 886657848, 2791106585, 8786123349, 27657838272, 87064092870, 274068969337, 862741412709, 2715822822365, 8549136056237, 26911817257385
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A011782, A034943, A109961, A369840, A373908.
Cf. A107025, A371125, A373905, A373906.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+4*k,n-2*k));

Formula

a(n) = 6*a(n-1) - 14*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: 1/(1 - x - x^2/(1 - x)^5).

A113066 Expansion of (1 + x)^2/((1 + x + x^2)*(1 + 3*x + x^2)).

Original entry on oeis.org

1, -2, 4, -10, 27, -72, 189, -494, 1292, -3382, 8855, -23184, 60697, -158906, 416020, -1089154, 2851443, -7465176, 19544085, -51167078, 133957148, -350704366, 918155951, -2403763488, 6293134513, -16475640050, 43133785636, -112925716858, 295643364939, -774004377960
Offset: 0

Views

Author

Creighton Dement, Oct 13 2005

Keywords

Comments

Binomial transform gives signed version of A093040.
The positive sequence has g.f. (1 - x)^2/((1 - x + x^2)(1 - 3*x + x^2)) and a(n) = Sum_{k=0..n} binomial(n+k+1, n-k)*(1+(-1)^k)/2. - Paul Barry, Jul 06 2009
Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'

References

  • C. Dement, Floretion Integer Sequences (work in progress).

Links

Crossrefs

Cf. A113067, A113068, A093040, A109961, A001906.

Programs

  • GAP
    a:=[1,-2,4,-10];; for n in [5..35] do a[n]:=-4*a[n-1]-5*a[n-2]-4*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 11 2018
  • Magma
    I:=[1,-2,4,-10]; [n le 4 select I[n] else -4*Self(n-1)-5*Self(n-2)- 4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 12 2018
  • Maple
    seq(coeff(series((1+x)^2/((1+x+x^2)*(1+3*x+x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Sep 11 2018
  • Mathematica
    LinearRecurrence[{-4, -5, -4, -1}, {1, -2, 4, -10}, 40] (* Vincenzo Librandi, Sep 12 2018 *)
CoefficientList[Series[(1 + x)^2/((1 + x + x^2)*(1 + 3 x + x^2)), {x, 0, 50}], x] (* Stefano Spezia, Sep 12 2018 *)
  • PARI
    x='x+O('x^99); Vec((1+x)^2/((1+x+x^2)*(1+3*x+x^2))) \\ Altug Alkan, Sep 11 2018

Formula

a(n) + a(n+1) = (-1)^(n+1)*A109961(n+1).
a(n) + a(n+1) + a(n+2) = (-1)^n*A001906(n+2) = (-1)^n*F(2*n+4).
a(n) = A049347(n)/2 + (-1)^n*A001906(n+1)/2. - R. J. Mathar, Nov 10 2009
Lim_{n -> inf} a(n)/a(n-1) = -(1 + A001622). - A.H.M. Smeets, Sep 11 2018
a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4). - Muniru A Asiru, Sep 11 2018
Showing 1-5 of 5 results.

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