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.

A099508 A transform of the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 9, 21, 25, 54, 68, 139, 183, 358, 489, 924, 1301, 2390, 3450, 6194, 9126, 16080, 24093, 41805, 63507, 108817, 167188, 283533, 439689, 739391, 1155384, 1929504, 3033991, 5038089, 7962758, 13161067, 20889012, 34394215, 54778872
Offset: 0

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Author

Paul Barry, Oct 20 2004

Keywords

Comments

A transform of A001045 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3)).

Links

Crossrefs

Cf. A099505.

Programs

  • Mathematica
    LinearRecurrence[{1,2,-2,1,0,-1},{0,1,1,3,3,8},40] (* Harvey P. Dale, Sep 25 2021 *)

Formula

G.f.: x/(1-x-2x^2+2x^3-x^4+x^6); a(n)=a(n-1)+2a(n-2)-2a(n-3)+a(n-4)-a(n-6); a(n)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*J(n-3k)}.

A099516 A transform of the Pell numbers.

Original entry on oeis.org

0, 1, 2, 5, 10, 23, 50, 112, 246, 545, 1202, 2658, 5870, 12972, 28656, 63315, 139880, 309049, 682790, 1508527, 3332850, 7363430, 16268356, 35942447, 79409300, 175442668, 387613604, 856372740, 1892023992, 4180136405, 9235369230
Offset: 0

Views

Author

Paul Barry, Oct 20 2004

Keywords

Comments

A transform of A000129 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3))

Links

Crossrefs

Cf. A099505, A099508.

Programs

  • Mathematica
    LinearRecurrence[{2,1,-2,2,0,-1},{0,1,2,5,10,23},40] (* Harvey P. Dale, Apr 15 2015 *)

Formula

G.f.: x/(1-2x-x^2+2x^3-2x^4+x^6); a(n)=2a(n-1)+a(n-2)-2a(n-3)+2a(n-4)-a(n-6); a(n)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*Pell(n-3k)}.

A099517 A transform of (1-x)/(1-2x).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 27, 44, 71, 116, 188, 305, 493, 798, 1291, 2090, 3382, 5473, 8855, 14328, 23183, 37512, 60696, 98209, 158905, 257114, 416019, 673134, 1089154, 1762289, 2851443, 4613732, 7465175, 12078908, 19544084, 31622993, 51167077
Offset: 0

Views

Author

Paul Barry, Oct 20 2004

Keywords

Comments

A transform of A011782 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3))

Links

Crossrefs

Cf. A099505, A099508, A099516, A000071.

Formula

G.f.: (1-x+x^3)/((1+x^3)*(1-2*x+x^3)).
a(n) = 2*a(n-1)-2*a(n-3)+2*a(n-4)-a(n-6).
a(n) = sum{k=0..floor(n/3), binomial(n-2*k, k)*(-1)^k*(2^(n-3*k)+0^(n-3*k))/2}.
a(n) = A057079(n+1)/6 +A000045(n+3)/2 -A010694(n)/6. - R. J. Mathar, Sep 21 2012
Showing 1-3 of 3 results.

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

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