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 1-4 of 4 results.

A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2).

Original entry on oeis.org

0, 0, 1, 3, 11, 35, 115, 371, 1203, 3891, 12595, 40755, 131891, 426803, 1381171, 4469555, 14463795, 46805811, 151466803, 490156851, 1586180915, 5132989235, 16610702131, 53753361203, 173949530931, 562912506675, 1821623137075, 5894896300851, 19076285150003
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It can be noticed that A014335/A011782 is an "autosequence", that is a sequence which is identical to its inverse binomial transform, except for alternating signs. - Jean-François Alcover, Jun 15 2016

Links

Crossrefs

Cf. A000032, A000045, A014334, A081057, A203579.
Cf. (partial sums of) A063727.
Column k=2 of A346415.

Programs

  • Magma
    [(2^n*Lucas(n)-2)/10: n in [0..40]]; // G. C. Greubel, Jan 06 2023
  • Maple
    a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]+1 od: seq(a[n], n=0..29); # Zerinvary Lajos, Dec 14 2008
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1,3]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 04 2016
  • Mathematica
    LinearRecurrence[{3,2,-4}, {0,0,1}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)
  • SageMath
    [(2^n*lucas_number2(n,1,-1) -2)/10 for n in range(41)] # G. C. Greubel, Jan 06 2023

Formula

a(n) = A014334(n)/2.
G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - Vladeta Jovovic, Mar 05 2003
E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - Vladeta Jovovic, Sep 01 2004
From Benoit Cloitre, Sep 25 2004: (Start)
a(n+1) = Sum_{i=0..n} A000045(i)*2^(i-1).
a(n) = (1/5)*(2^(n-1)*A000032(n) - 1). (End)
a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - Zerinvary Lajos, Dec 14 2008
G.f.: G(0)*x^2/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = (A203579(n) - 2)/5. - Vladimir Reshetnikov, Oct 06 2016

A256180 Exponential transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 21, 86, 404, 2121, 12264, 77272, 525941, 3839706, 29891370, 246906569, 2154904856, 19799299506, 190904273049, 1926229186162, 20288311652672, 222568337565537, 2537998989244956, 30029233006187756, 368050599579654557, 4665833729558724030
Offset: 0

Views

Author

Alois P. Heinz, Mar 18 2015

Keywords

Links

Crossrefs

Cf. A000045, A006701, A007552, A112005.
Row sums of A346415.

Programs

  • Maple
    F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *F(j) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    Table[Sum[BellY[n, k, Fibonacci[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Formula

E.g.f: exp(1/sqrt(5)*(exp((1+sqrt(5))*x/2)-exp((1-sqrt(5))*x/2))).
E.g.f: exp(2/5*sqrt(5)*exp(x/2)*sinh(sqrt(5)*x/2)).

A014337 Three-fold exponential convolution of Fibonacci numbers with themselves (divided by 6).

Original entry on oeis.org

0, 0, 0, 1, 6, 35, 180, 910, 4494, 22049, 107580, 523765, 2546280, 12370436, 60074196, 291677905, 1416019290, 6873991055, 33368325084, 161976597634, 786259649010, 3816607413905, 18526261416960
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A014336.
Column k=3 of A346415.

Programs

  • Mathematica
    CoefficientList[Series[x^3/((1-3*x+x^2)*(1-3*x-9*x^2)),{x,0,30}],x] (* Vincenzo Librandi, Mar 20 2012 *)
  • PARI
    a(n)=(3^n*fibonacci(n)-3*fibonacci(2*n))/30 \\ Charles R Greathouse IV, Mar 20 2012

Formula

a(n) = (1/30)(3^n*Fibonacci(n) - 3*Fibonacci(2n)). - Ralf Stephan, May 14 2004
G.f.: x^3/((1-3*x+x^2)*(1-3*x-9*x^2)). - Colin Barker, Mar 19 2012

A014341 Four-fold exponential convolution of Fibonacci numbers with themselves (divided by 24).

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 85, 630, 4445, 30282, 202565, 1338590, 8781245, 57330130, 373158149, 2424055270, 15726655645, 101945272410, 660482562565, 4277608337518, 27697444314045, 179313224367970, 1160758151567685, 7513510490678390, 48632383711996701, 314772014292259050
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A014340.
Column k=4 of A346415.

Formula

G.f.: -x^4/((2*x-1)*(16*x^2+4*x-1)*(x^2+4*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
Showing 1-4 of 4 results.

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

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