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 1-3 of 3 results.

A167993 Expansion of x^2/((3*x-1)*(3*x^2-1)).

Original entry on oeis.org

0, 0, 1, 3, 12, 36, 117, 351, 1080, 3240, 9801, 29403, 88452, 265356, 796797, 2390391, 7173360, 21520080, 64566801, 193700403, 581120892, 1743362676, 5230147077, 15690441231, 47071500840, 141214502520, 423644039001, 1270932117003, 3812797945332, 11438393835996
Offset: 0

Views

Author

Paul Curtz, Nov 16 2009

Keywords

Comments

The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]

Links

Crossrefs

Cf. A138587, A107767 (partial sums).

Programs

  • Mathematica
    CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)
LinearRecurrence[{3,3,-9},{0,0,1},30] (* Harvey P. Dale, Nov 05 2017 *)
  • PARI
    Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 29 2011

Formula

a(2*n+1) = 3*a(2*n).
a(2*n) = A122006(2*n)/2.
a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
a(n+1) - a(n) = A122006(n).
a(n) = (3^n - A108411(n+1))/6.
G.f.: x^2/((3*x-1)*(3*x^2-1)).
From Colin Barker, Sep 23 2016: (Start)
a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.
a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.
(End)

Extensions

Formulae corrected by Johannes W. Meijer, Jun 28 2011

A135098 Duplicate of A136488.

Original entry on oeis.org

1, 2, 5, 10, 22, 44, 92, 184, 376, 752, 1520, 3040, 6112, 12224, 24512, 49024, 98176, 196352, 392960, 785920, 1572352, 3144704, 6290432, 12580864, 25163776, 50327552, 100659200, 201318400, 402644992, 805289984, 1610596352, 3221192704
Offset: 0

Views

Author

Paul Curtz, Feb 12 2008

Keywords

Comments

Previous name was: First differences of A135094.
Apart to offset same as A136488.

Links

Crossrefs

Cf. A135094, A136488 (same up to offset).

Programs

  • Mathematica
    Table[2^((n - 5)/2)*( 3*2^((n + 1)/2) - (1 - (-1)^n) - (1 + (-1)^n)*Sqrt[2] ), {n, 1, 50}] (* or *) LinearRecurrence[{2, 2, -4}, {1, 2, 5}, 25] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; -4,2,2]^n*[1;2;5])[1,1] \\ Charles R Greathouse IV, Sep 23 2016

Formula

From R. J. Mathar, Feb 15 2008: (Start)
O.g.f.: (2*x+1) / (2*(2*x^2-1)) -3 / (2*(2*x-1)).
a(n) = (-A016116(n+1) +A007283(n)) / 2 . (End)
G.f.: (1 - x)*(1 + x) / ((1 - 2*x)*(1 - 2*x^2)). - Arkadiusz Wesolowski, Oct 24 2013
From G. C. Greubel, Sep 23 2016: (Start)
a(n) = 2^((n-4)/2)*( 6*2^(n/2) - (1 + (-1)^n) - (1 - (-1)^n)*sqrt(2) ).
E.g.f.: (1/2)*( 3*exp(2*x) - cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x) ). (End) [corrected by Jason Yuen, Sep 25 2024]
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Wesley Ivan Hurt, Apr 07 2021

Extensions

More terms from R. J. Mathar, Feb 15 2008

A342762 Fold a square sheet of paper alternately vertically to the left and horizontally downwards; after each fold, draw a line along each inward crease; after n folds, the resulting graph has a(n) connected components.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 10, 20, 42, 86, 178, 362, 738, 1490, 3010, 6050, 12162, 24386, 48898, 97922, 196098, 392450, 785410, 1571330, 3143682, 6288386
Offset: 0

Views

Author

Rémy Sigrist and N. J. A. Sloane, Mar 21 2021

Keywords

Comments

A342759 is the main sequence for this entry.

Examples

			See illustration in Links section.

Links

Crossrefs

Cf. A342759.
It appears that a(n) = A135094(n-4) + 2 for n >= 5. Hugo Pfoertner, Mar 29 2021
Showing 1-3 of 3 results.

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