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.

A094707 Partial sums of repeated Fibonacci sequence.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 32, 40, 53, 66, 87, 108, 142, 176, 231, 286, 375, 464, 608, 752, 985, 1218, 1595, 1972, 2582, 3192, 4179, 5166, 6763, 8360, 10944, 13528, 17709, 21890, 28655, 35420, 46366, 57312, 75023, 92734, 121391, 150048, 196416
Offset: 0

Views

Author

Paul Barry, May 21 2004

Keywords

Comments

Equals row sums of triangle A139147 starting with "1". - Gary W. Adamson, Apr 11 2008

Links

Crossrefs

Cf. A000045, A103609, A131524, A139147.

Programs

  • Magma
    [Fibonacci(Floor((n+6)/2))*((n+1) mod 2) + 2*Fibonacci(Floor((n+3)/2))*(n mod 2) - 2: n in [0..60]]; // G. C. Greubel, Feb 12 2023
  • Mathematica
    LinearRecurrence[{1,1,-1,1,-1}, {0,0,1,2,3}, 50] (* Jean-François Alcover, Nov 18 2017 *)
  • SageMath
    def A094707(n): return fibonacci((n+6)//2) - 2 if (n%2==0) else 2*fibonacci((n+3)//2) - 2
[A094707(n) for n in range(61)] # G. C. Greubel, Feb 12 2023

Formula

G.f. : x^2*(1+x)/((1-x)*(1-x^2-x^4)).
a(n) = a(n-1) + a(n+2) - a(n-3) + a(n-4) - a(n-5).
a(n) = Sum_{k=0..n} Fibonacci(floor(k/2)).
a(n) = -2 - (sqrt(5)/2 - 1/2)^(n/2)*((2*sqrt(5)/5 - 1)*cos(Pi*n/2) + sqrt(4*sqrt(5)/5 - 8/5)*sin(Pi*n/2)) - (sqrt(5)/2 + 1/2)^(n/2)*((sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5 - 1/2)*(-1)^n - sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5-1/2).
a(n) = A131524(n) + A131524(n+1). - R. J. Mathar, Jul 07 2011
a(n) = Fibonacci(n/2 +3) - 2 if n even, otherwise a(n) = 2*Fibonacci((n-1)/2 + 2) - 2. - G. C. Greubel, Feb 12 2023

A139038 Triangle read by rows: T(n,m) = A000931(m+6) if m <= floor(n/2), A000931(n+6-m) otherwise, for 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 4, 3, 2, 2, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, May 31 2008

Keywords

Comments

The Padovan sequence is pushed back to a(-1)=1, so that the triangle is not almost all ones.

Examples

			The triangle begins:
  1;
  1, 1;
  1, 1, 1;,
  1, 1, 1, 1;
  1, 1, 2, 1, 1;
  1, 1, 2, 2, 1, 1;
  1, 1, 2, 2, 2, 1, 1;
  1, 1, 2, 2, 2, 2, 1, 1;
  1, 1, 2, 2, 3, 2, 2, 1, 1;
  1, 1, 2, 2, 3, 3, 2, 2, 1, 1;
  1, 1, 2, 2, 3, 4, 3, 2, 2, 1, 1;
  ...

Crossrefs

Cf. A000931, A139147.

Programs

  • Mathematica
    a[-1] = 1; a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; (* Padovan : A000931 *)
Table[If[m <= Floor[n/2], a[m], a[n - m]], {n, 0, 10}, {m, 0, n}]

Extensions

Edited by N. J. A. Sloane, Feb 28 2009
Non-ASCII characters in %t line corrected by Wouter Meeussen, Feb 10 2013
Definition corrected and offset changed by Georg Fischer, May 16 2024

A139040 Triangle read by rows: each row is an initial segment of the terms of A000930 followed by its reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 9, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 9, 9, 6, 4, 3, 2, 1, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, May 31 2008

Keywords

Examples

			Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{1, 1, 1, 1},
{1, 1, 2, 1, 1},
{1, 1, 2, 2, 1, 1},
{1, 1, 2, 3, 2, 1, 1},
{1, 1, 2, 3, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 4, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 6, 4, 3, 2, 1, 1}

Links

Crossrefs

Cf. A139147, A000930. Row sums are in A238383.

Programs

  • Maple
    A000930 := proc(n) coeftayl( 1/(1-x-x^3),x=0,n) ; end: A139040 := proc(n,m) A000930(min(m,n+1-m)) ; end: for n from 1 to 16 do for m from 1 to n do printf("%d,",A139040(n,m)) ; od: od: # R. J. Mathar, Jun 08 2008
  • Mathematica
    a[-2]=0;a[-1]=1;a[0]=1;a[n_]:=a[n]=a[n-1]+a[n-3];(*A000930*)
g[n_,m_]:=If[m <= Floor[n/2],a[m],a[n-m]];w=Table[Table[g[n,m],{m,0,n}],{n,0,10}]; Flatten[w]

Extensions

Edited and corrected by N. J. A. Sloane, Jun 30 2008
Corrected by Philippe Deléham, Feb 25 2014

A139039 A triangular central symmetric sequence based on the sequence A003269: if m <= floor(n/2), t(n,m) = A003269(m+2), otherwise t(n,m) = A003269(n - (m+2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, May 31 2008

Keywords

Comments

Row sums: {1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, ...}. [Is this A186445 or A080078? - N. J. A. Sloane, Feb 10 2013]
The A003269 sequence is pushed back twice, so that the triangle is not almost all ones.

Examples

			{1},
{1, 1},
{1, 1, 1},
{1, 1, 1, 1},
{1, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1},
{1, 1, 1, 2, 1, 1, 1},
{1, 1, 1, 2, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 3, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1}

Crossrefs

Cf. A139147, A003269.

Programs

  • Mathematica
    Clear[a]; a[ -2] = 0; a[ -1] = 1; a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 4]; (* A003269 *) Table[If[m <= Floor[n/2],a[m],a[n-m] ] ,{n,0,10},{m,0,n}]

Formula

a(n) = a(n-1) + a(n-4); t(n,m) = a(m) if m <= floor(n/2), a(n-m) otherwise.

Extensions

Non-ASCII characters removed and Mathematica code corrected by Wouter Meeussen, Feb 10 2013
Showing 1-4 of 4 results.

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

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