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.

A117915 Erroneous version of A117501.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 4, 5, 5, 1, 5, 5, 7, 8, 8, 1, 6, 6, 9, 11, 13, 13, 1, 7, 7, 11, 14, 18, 21, 21, 1, 8, 8, 13, 27, 23, 29, 34, 34, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55
Offset: 1

Views

Author

Keywords

A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 3, 0, 1, 4, 4, 5, 5, 0, 1, 5, 5, 7, 8, 8, 0, 1, 6, 6, 9, 11, 13, 13, 0, 1, 7, 7, 11, 14, 18, 21, 21, 0, 1, 8, 8, 13, 17, 23, 29, 34, 34, 0, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 0, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89
Offset: 0

Views

Author

Ross La Haye, Aug 11 2005; corrected Apr 14 2006

Keywords

Comments

Lower triangular version is at A117501. - Ross La Haye, Apr 12 2006

Examples

			Table starts:
[0] 0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ...
[1] 0, 1,  2,  3,  5,  8, 13, 21,  34,  55, ...
[2] 0, 1,  3,  4,  7, 11, 18, 29,  47,  76, ...
[3] 0, 1,  4,  5,  9, 14, 23, 37,  60,  97, ...
[4] 0, 1,  5,  6, 11, 17, 28, 45,  73, 118, ...
[5] 0, 1,  6,  7, 13, 20, 33, 53,  86, 139, ...
[6] 0, 1,  7,  8, 15, 23, 38, 61,  99, 160, ...
[7] 0, 1,  8,  9, 17, 26, 43, 69, 112, 181, ...
[8] 0, 1,  9, 10, 19, 29, 48, 77, 125, 202, ...
[9] 0, 1, 10, 11, 21, 32, 53, 85, 138, 223, ...

Links

Crossrefs

Rows: A000045(j); A000045(j+1), for j > 0; A000032(j), for j > 0; A000285(j-1), for j > 0; A022095(j-1), for j > 0; A022096(j-1), for j > 0; A022097(j-1), for j > 0. Diagonals: a(i, i) = A094588(i); a(i, i+1) = A007502(i+1); a(i, i+2) = A088209(i); Sum[a(i-j, j), {j=0...i}] = A104161(i). a(i, j) = A101220(i, 0, j).
Rows 7 - 19: A022098(j-1), for j > 0; A022099(j-1), for j > 0; A022100(j-1), for j > 0; A022101(j-1), for j > 0; A022102(j-1), for j > 0; A022103(j-1), for j > 0; A022104(j-1), for j > 0; A022106(j-1), for j > 0; A022107(j-1), for j > 0; A022108(j-1), for j > 0; A022109(j-1), for j > 0; A022110(j-1), for j > 0.
a(2^i-2, j+1) = A118654(i, j), for i > 0.
Cf. A117501.

Programs

  • Maple
    A := (n, k) -> ifelse(k = 0, 0,
      n*combinat:-fibonacci(k-1) + combinat:-fibonacci(k)):
seq(seq(A(n - k, k), k = 0..n), n = 0..6); # Peter Luschny, May 28 2022
  • Mathematica
    T[n_, 0]:= 0; T[n_, 1]:= 1; T[n_, 2]:= n - 1; T[n_, 3]:= n - 1; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)

Formula

a(i, 0) = 0, a(i, j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0.
a(i, 0) = 0, a(i, 1) = 1, a(i, 2) = i+1, a(i, j) = a(i, j-1) + a(i, j-2), for j > 2.
G.f.: (x*(1 + ix))/(1 - x - x^2).
Sum_{j=0..i+1} a(i-j+1, j) - Sum_{j=0..i} a(i-j, j) = A001595(i). - Ross La Haye, Jun 03 2006

Extensions

More terms from G. C. Greubel, Jan 07 2017

A104161 G.f.: x*(1 - x + x^2)/((1-x)^2 * (1 - x - x^2)).

Original entry on oeis.org

0, 1, 2, 5, 10, 19, 34, 59, 100, 167, 276, 453, 740, 1205, 1958, 3177, 5150, 8343, 13510, 21871, 35400, 57291, 92712, 150025, 242760, 392809, 635594, 1028429, 1664050, 2692507, 4356586, 7049123
Offset: 0

Views

Author

Creighton Dement, Mar 10 2005

Keywords

Comments

A floretion-generated sequence.
Floretion Algebra Multiplication Program, FAMP Code: 1vesrokseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)('i + i' + 'ji' + 'ki' + e) ] RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + p.
Partial sums of Leonardo numbers A001595. - Jonathan Vos Post, Jan 01 2011

Links

Crossrefs

Cf. A000045, A006355, A001595, A117501, A145912.

Programs

  • GAP
    List([0..40], n-> 2*Fibonacci(n+2) -(n+2)); # G. C. Greubel, Jul 09 2019
  • Magma
    [2*Fibonacci(n+2) -(n+2): n in [0..40]]; // G. C. Greubel, Jul 09 2019
  • Mathematica
    a=0;b=1;Table[c=b+a+n; a=b; b=c, {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[x*(1-x+x^2)/((1-x)^2*(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,2,5},40] (* Harvey P. Dale, Sep 06 2012 *)
  • PARI
    my(x='x+O('x^40)); concat(0, Vec(x*(1-x+x^2)/((1-x)^2*(1-x-x^2)))) \\ G. C. Greubel, Sep 26 2017
  • SageMath
    [2*fibonacci(n+2) -(n+2) for n in (0..40)] # G. C. Greubel, Jul 09 2019

Formula

Superseeker results (incomplete): a(2) - 2a(n+1) + a(n) = A006355(n+1) (Number of binary vectors of length n containing no singletons); a(n+1) - a(n) = A001595(n) (2-ranks of difference sets constructed from Segre hyperovals); a(n) + n + 1 = A001595(n+1).
A107909(a(n)) = A000975(n). - Reinhard Zumkeller, May 28 2005
From Ross La Haye, Aug 03 2005: (Start)
a(n) = 2*(Fibonacci(n+2) - 1) - n.
a(n) = Sum_{k=0..n} A101220(n-k, 0, k). (End)
From Gary W. Adamson, Apr 02 2006: (Start)
a(n) = a(n-1) + a(n-2) + n-1.
a(n) = row sums of A117501, starting (1, 2, 5, 10, ...). (End)
a(n) = Sum_{k=0..n} A109754(n-k,k). - Ross La Haye, Apr 12 2006
a(n) = (Sum_{k=0..n} (n-k)*Fibonacci(k-1) + Fibonacci(k)) - n. - Ross La Haye, May 31 2006
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = -2 - n + (-A094214)^n*(1-A010499/5) + (1+A010499/5)/A094214^n.
a(n) = A006355(n+3) - n - 2. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5. - Harvey P. Dale, Sep 06 2012

A117502 Triangle, row sums = A001595.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 2, 5, 1, 1, 2, 3, 8, 1, 1, 2, 3, 5, 13, 1, 1, 2, 3, 5, 8, 21, 1, 1, 2, 3, 5, 8, 13, 34, 1, 1, 2, 3, 5, 8, 13, 21, 55, 1, 1, 2, 3, 5, 8, 13, 21, 34, 89, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 144, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 233
Offset: 1

Views

Author

Gary W. Adamson, Mar 23 2006

Keywords

Comments

Row sums = A001595 = (1, 3, 9, 15, 25, 41, ...).

Examples

			Row 5 of the triangle = (1, 1, 2, 3, 8); the first 5 Fibonacci terms with a deletion of F(5) = 5.
First few rows of the triangle are:
  1;
  1, 2;
  1, 1, 3;
  1, 1, 2, 5;
  1, 1, 2, 3, 8; ...

Links

Crossrefs

Cf. A000045, A117501, A001595.

Programs

  • GAP
    T:= function(n,k)
    if k=n then return Fibonacci(n+1);
    else return Fibonacci(k);
    fi;
  end;
Flat(List([1..20], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Jul 14 2019
  • Magma
    [k eq n select Fibonacci(n+1) else Fibonacci(k): k in [1..n], n in [1..20]]; // G. C. Greubel, Jul 10 2019
  • Mathematica
    Table[If[k==n, Fibonacci[n+1], Fibonacci[k]], {n, 20}, {k, n}]//Flatten (* G. C. Greubel, Jul 10 2019 *)
  • PARI
    T(n,k) = if(k==n, fibonacci(n+1), fibonacci(k)); \\ G. C. Greubel, Jul 10 2019
  • Sage
    def T(n, k):
    if (k==n): return fibonacci(n+1)
    else: return fibonacci(k)
[[T(n, k) for k in (1..n)] for n in (1..20)] # G. C. Greubel, Jul 10 2019

Formula

n-th row = first n Fibonacci terms, with a deletion of F(n).
Columns of the triangle are difference terms of the array in A117501.
T(n,k) = Fibonacci(k) for k < n and T(n,n) = Fibonacci(n+1). - G. C. Greubel, Jul 10 2019

Extensions

More terms added by G. C. Greubel, Jul 10 2019
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.