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 1-2 of 2 results.

A078017 Expansion of (1-x)/(1-x+x^2+2*x^3).

Original entry on oeis.org

1, 0, -1, -3, -2, 3, 11, 12, -5, -39, -58, -9, 127, 252, 143, -363, -1010, -933, 803, 3756, 4819, -543, -12874, -21969, -8009, 39708, 91655, 67965, -103106, -354381, -387205, 173388, 1269355, 1870377, 254246, -4154841, -8149841, -4503492, 11956031, 32759205, 29810158, -26861109
Offset: 0

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Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077952.

Programs

  • GAP
    a:=[1,0,-1];; for n in [4..50] do a[n]:=a[n-1]-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 29 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x+x^2+2*x^3) )); // G. C. Greubel, Jun 29 2019
  • Mathematica
    LinearRecurrence[{1,-1,-2}, {1,0,-1}, 50] (* or *) CoefficientList[ Series[(1-x)/(1-x+x^2+2*x^3), {x,0,50}], x] (* G. C. Greubel, Jun 29 2019 *)
  • PARI
    Vec((1-x)/(1-x+x^2+2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 27 2012
  • Sage
    ((1-x)/(1-x+x^2+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

Formula

G.f.: (1-x)/(1-x+x^2+2*x^3).
a(n) = A077952(n) - A077952(n-1). - G. C. Greubel, Jun 29 2019

A102785 G.f.: (x-1)/(-2*x^2 + 3*x^3 + 2*x - 1).

Original entry on oeis.org

1, 1, 0, 1, 5, 8, 9, 17, 40, 73, 117, 208, 401, 737, 1296, 2321, 4261, 7768, 13977, 25201, 45752, 83033, 150165, 271520, 491809, 891073, 1613088, 2919457, 5285957, 9572264, 17330985, 31375313, 56805448
Offset: 0

Views

Author

Creighton Dement, Feb 11 2005

Keywords

Comments

Inverse binomial transform of A078017. Inversion of A052102.
Floretion Algebra Multiplication Program, FAMP Code: 4jbasekseq[ (+ 'ii' + 'jj' + 'ij' + 'ji' + e)*x) ] where x is defined as 1/4 times the sum of all 16 floretion basis vectors.

Links

Crossrefs

Cf. A078017, A052102, A077952.

Programs

  • Maxima
    a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j),j,0,k))*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* Vladimir Kruchinin, May 05 2011 */
  • Maxima
    makelist(coeff(taylor((x-1)/(-2*x^2+3*x^3+2*x-1), x, 0, n), x, n), n, 0, 32); /* Bruno Berselli, May 30 2011 */

Formula

a(n+3) = 2a(n+2) - 2a(n+1) + 3a(n), a(0) = 1, a(1) = 1, a(2) = 0
a(n) = Sum(k=1..n, Sum(i=k..n, (Sum(j=0..k, binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j)))*binomial(n+k-i-1,k-1))), n > 0, a(0)=1. - Vladimir Kruchinin, May 05 2011
Showing 1-2 of 2 results.

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

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