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.

A117189 Binomial transform of the tribonacci sequence A000073 (shifted left twice).

Original entry on oeis.org

1, 2, 5, 14, 40, 114, 324, 920, 2612, 7416, 21056, 59784, 169744, 481952, 1368400, 3885280, 11031424, 31321376, 88930368, 252498816, 716916544, 2035531648, 5779458048, 16409538688, 46591385856, 132286304768, 375598753024, 1066432564736, 3027907856384
Offset: 0

Views

Author

Gary W. Adamson, Mar 01 2006

Keywords

Comments

a(n)/a(n-1) tends to 2.83928675... = A058265 + 1.
Partial sums are in A073357. - R. J. Mathar, Apr 02 2008

Examples

			a(4) = 14 = 1*1 + 3*1 + 3*2 + 1*4;
a(6) = 324 = 2*114 + 1*40 + 2*14 + 3*5 + 4*2 + 5*1. - _Bob Selcoe_, Jun 28 2014

Links

Crossrefs

Cf. A000073, A115390.

Programs

  • Mathematica
    CoefficientList[Series[-(x - 1)^2/(-1 + 4*x - 4*x^2 + 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 05 2014 *)
LinearRecurrence[{4,-4,2},{1,2,5},40] (* Harvey P. Dale, Oct 10 2016 *)

Formula

Binomial transform of A000073 starting with A000073(2): (1, 1, 2, 4, 7, 13, ...).
a(n) = 4*a(n-1)-4*a(n-2)+2*a(n-3), n>2. - T. D. Noe, Nov 07 2006
O.g.f.: -(x-1)^2/(-1+4*x-4*x^2+2*x^3). - R. J. Mathar, Apr 02 2008
a(n) = 2*a(n-1) + Sum_{j=1..n-1} j*a(n-j-1), n>=1; with a(0) = 1. - Bob Selcoe, Jun 28 2014

Extensions

Corrected and extended by T. D. Noe, Nov 07 2006

A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 147, 419, 1191, 3383, 9607, 27279, 77455, 219919, 624415, 1772895, 5033759, 14292287, 40579903, 115217983, 327136895, 928835455, 2637230207, 7487852799, 21260161279, 60363694335, 171389837823, 486624896511, 1381667623423, 3922950583295
Offset: 0

Views

Author

Roger L. Bagula, Mar 26 2005

Keywords

Comments

Examples of the morphism starting with {1} are shown in A103684. Counting the total number of '1' in rows 1 to n of A103684 yields 1, 3, 8,... = A073357(n+1),
counting the total number of '2' in rows 1 to n yields 0, 1, 4,.. = A115390(n+1),
and counting the total number '3' in rows 1 to n yields a(n), the sequence here.
Inverse binomial transform yields 0, 0, 1, 2, 3, 6, 11, 20,..., a variant of A001590 [Nov 18 2010]

Links

Crossrefs

Cf. A073058, A103684.

Programs

  • Mathematica
    LinearRecurrence[{5,-8,6,-2},{0,0,1,5},30] (* Harvey P. Dale, Nov 10 2011 *)

Formula

a(n)= +5*a(n-1) -8*a(n-2) +6*a(n-3) -2*a(n-4) = a(n-1)+A115390(n). [Nov 18 2010]
G.f.: x^2 / ( (x-1)*(2*x^3-4*x^2+4*x-1) ). [Nov 18 2010]

Extensions

Depleted by the information already in A073357 and A115390; corrected image of {2} in the defn. - The Assoc. Eds. of the OEIS, Nov 18 2010

A073358 Binomial transform, alternating in sign, of the tribonacci numbers.

Original entry on oeis.org

0, -1, -1, -2, -2, -2, 0, 4, 12, 24, 40, 56, 64, 48, -16, -160, -416, -800, -1280, -1728, -1856, -1152, 1152, 6016, 14336, 26368, 40704, 52736, 52736, 24064, -57344, -220160, -488448, -862208, -1284096, -1591296, -1458176, -348160
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Jul 30 2002

Keywords

References

  • Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Links

Crossrefs

Cf. A000073, A073357.

Programs

  • Mathematica
    h[n_] := h[n]=2*h[n-1]-2*h[n-3]; h[0]=0; h[1]=-1; h[2]=-1

Formula

a(n)=2*a(n-1)-2*a(n-3), a(0)=0, a(1)=-1, a(2)=-1. Generating function A(x)=x*(x-1)/(1-2x+2x^3).
Showing 1-3 of 3 results.

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