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.

A301699 Generating function = g(g(x)), where g(x) = g.f. of Jacobsthal numbers A001045.

Original entry on oeis.org

0, 1, 2, 8, 26, 94, 330, 1178, 4186, 14914, 53098, 189122, 673530, 2398834, 8543498, 30428162, 108371354, 385970386, 1374653610, 4895901602, 17437011514, 62102837746, 221182535242, 787753281218, 2805624912090, 9992381298706, 35588393716202
Offset: 0

Views

Author

N. J. A. Sloane, Mar 29 2018

Keywords

Comments

The Dira (2017) article describes this as the self-convolution of A001045, but it is really the self-composition. - N. J. A. Sloane, Apr 07 2019, following a suggestion from Ilya Gutkovskiy. Note that A073371 is the convolution of A001045(n+1) with itself, with g.f.: g(x)^2/x^2, where g(x) = g.f. of A001045.
The Dira (2017) article contains on pages 851 and 852 several other sequences that could be added to the OEIS.

Links

Crossrefs

Cf. A001045, A073371, A270863.

Programs

  • Magma
    I:=[0,1,2,8]; [n le 4 select I[n] else 3*Self(n-1)+4*Self(n-2)-6*Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 30 2018
  • Maple
    f:=proc(a,b) local t1;
t1:=(x-a*x^2-b*x^3)/(1-3*a*x+(2*a^2-3*b)*x^2+3*a*b*x^3 + b^2*x^4);
lprint(t1);
series(t1,x,50);
seriestolist(%);
end;
f(1,2);
  • Mathematica
    CoefficientList[Series[(-2 x^3 - x^2 + x) / (4 x^4 + 6 x^3 - 4 x^2 - 3 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 30 2018 *)

Formula

G.f.: (-2*x^3-x^2+x)/(4*x^4+6*x^3-4*x^2-3*x+1).
a(n) = 3*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4). - Vincenzo Librandi, Mar 30 2018

A279277 Composition of Lucas numbers A000032 with Fibonacci numbers A000045.

Original entry on oeis.org

1, 4, 12, 37, 110, 327, 968, 2864, 8469, 25040, 74029, 218856, 647008, 1912753, 5654670, 16716883, 49420052, 146100276, 431915561, 1276869920, 3774804441, 11159436284, 32990587972, 97529916957, 288327225550, 852380393407, 2519888066928, 7449533000584, 22023018662909
Offset: 1

Views

Author

Oboifeng Dira, Dec 10 2016

Keywords

Comments

G(F(x)) where F(x) = x+x^2+2x^3+3x^4+... is the generating series of the Fibonacci numbers A000045 and G(x) = x+3x^2+4x^3+7x^4 +... is the generating series of the Lucas numbers A000032.

Examples

			(x+x^2)/(1-3x) = x + (3+1)x^2+... so a(1) = 1 and a(2) = 4.

Links

Crossrefs

Cf. A000045, A000032, A270863.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[(x + x^2 - x^3)/(1 - 3 x - x^2 + 3 x^3 + x^4), {x, 0, 24}], x] (* Michael De Vlieger, Dec 12 2016 *)

Formula

G.f. x*(1+x-x^2)/(1-3*x-x^2+3*x^3+x^4).
a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4), a(1)=1, a(2)=4, a(3)=12, a(4)=46.

A279285 Self-composition of the Pell numbers; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000129.

Original entry on oeis.org

0, 1, 4, 18, 82, 377, 1740, 8045, 37226, 172314, 797744, 3693493, 17101128, 79180525, 366618808, 1697509962, 7859781454, 36392245541, 168502887396, 780199897985, 3612471696230, 16726421117538, 77446465948772, 358591660029577, 1660346632032144, 7687716275234809, 35595568065121900, 164814155562334914
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 09 2016

Keywords

Links

Crossrefs

Cf. A000129, A270863.

Programs

  • Mathematica
    CoefficientList[Series[x (1 - 2 x - x^2)/(1 - 6 x + 5 x^2 + 6 x^3 + x^4), {x, 0, 27}], x]
LinearRecurrence[{6, -5, -6, -1}, {0, 1, 4, 18}, 28]

Formula

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

A286012 A Kedlaya-Wilf matrix for the Fibonacci sequence A000045.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 17, 5, 1, 5, 20, 48, 50, 8, 1, 6, 30, 102, 197, 147, 13, 1, 7, 42, 185, 532, 815, 434, 21, 1, 8, 56, 303, 1165, 2804, 3391, 1282, 34
Offset: 1

Views

Author

Oboifeng Dira, Apr 30 2017

Keywords

Comments

For any power series f(x) starting with the term x the first column of the Kedlaya-Wilf matrix are the coefficients of f(x), the second column are the coefficients of f(f(x)), the third column are the coefficients of f(f(f(x))) and so on. This gives a matrix with first row consisting of ones. The sequence given is the diagonal reading of this matrix from right up to left down.

Examples

			f^(3)(x) = x + 3x^2 + 12x^4 + ... as in A283679, so a(4)=1, a(8)=3, a(13)=12.

Links

Crossrefs

Cf. A000045, A270863, A283679.

Programs

  • Maple
    h:= x-> x/(1-x-x^2):
h2:= n-> coeff(series(h(h(x))), x, n+1), x, n):
h3:= n -> coeff(series(h(h2(x))),x,n+1), x, n):
etc.
h7:= n -> coeff(series(h(h6(x))),x,n+1), x, n): N7:=array(1..7,1..7,sparse): gg:=array([h1,h2,h3,h4,h5,h6,h7]):for k from 1 to 7 do: for j from 1 to 7 do: N7[k,j]:=coeff(series(gg[j],x,12),x^k): od:od:

Formula

As an n X n matrix a(i,j) = coefficient of x^i in f^(j)(x) for i,j=1..n where f^(j) is the j-fold composition of f with itself.

A302357 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of Fibonacci numbers (A000045).

Original entry on oeis.org

1, 2, 12, 102, 1165, 16603, 283283, 5624556, 127309302, 3234191224, 91094448874, 2816800580606, 94848640788603, 3454303753062123, 135278798460362984, 5668566821430630300, 253050028467629998389, 11988740253545762393562
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of g.f. A(x) = x/(1 - x - x^2) are as follows:
n = 1: 0, (1), 1,   2,    3,     5,  ... g.f. A(x)
n = 2: 0,  1, (2),  6,   17,    50,  ... g.f. A(A(x))
n = 3: 0,  1,  3, (12),  48,   197,  ... g.f. A(A(A(x)))
n = 4: 0,  1,  4,  20, (102),  532,  ... g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  30,  185, (1165), ... g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000045, A119821, A158832, A270863, A283679.

Programs

  • Mathematica
    Table[SeriesCoefficient[Nest[Function[x, x/(1 - x - x^2)], x, n], {x, 0, n}], {n, 18}]
Showing 1-5 of 5 results.

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

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