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.

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A131411 Triangle read by rows: T(n,k) = Fibonacci(n) + Fibonacci(k) - 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 9, 8, 8, 9, 10, 12, 15, 13, 13, 14, 15, 17, 20, 25, 21, 21, 22, 23, 25, 28, 33, 41, 34, 34, 35, 36, 38, 41, 46, 54, 67, 55, 55, 56, 57, 59, 62, 67, 75, 88, 109, 89, 89, 90, 91, 93, 96, 101, 109, 122, 143, 177, 144, 144, 145, 146, 148, 151, 156, 164, 177, 198, 232, 287
Offset: 1

Views

Author

Gary W. Adamson, Jul 08 2007

Keywords

Comments

Left column = Fibonacci numbers. Right column = A001595: (1, 1, 3, 5, 9, 15, 25,...).
Row sums = A131412: (1, 2, 7, 15, 32, 62, 117, 214,...).

Examples

			First few rows of the triangle are:
   1;
   1,  1;
   2,  2,  3;
   3,  3,  4,  5;
   5,  5,  6,  7,  9;
   8,  8,  9, 10, 12, 15;
  13, 13, 14, 15, 17, 20, 25;
  21, 21, 22, 23, 25, 28, 33, 41;
  ...

Links

Crossrefs

Row sums are A131412.
Cf. A000012, A000045, A001595, A104763, A127647, A131410.

Programs

  • GAP
    F:=Fibonacci;; Flat(List([1..15], n-> List([1..n], k-> F(n) +F(k) -1 ))); # G. C. Greubel, Jul 13 2019
  • Magma
    F:=Fibonacci; [F(n)+F(k)-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019
  • Mathematica
    With[{F=Fibonacci}, Table[F[n]+F[k]-1, {n,15}, {k,n}]//Flatten] (* G. C. Greubel, Jul 13 2019 *)
  • PARI
    T(n,k) = if(k<=n, fibonacci(n) + fibonacci(k) - 1, 0); \\ Andrew Howroyd, Aug 10 2018
  • Sage
    f=fibonacci; [[f(n)+f(k)-1 for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

Formula

Equals A131410 + A104763 - A000012 as infinite lower triangular matrices.

Extensions

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A104733 Triangle T(n,k) = sum_{j=k..n} Fibonacci(n-j+1)*Fibonacci(k+1), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 7, 4, 4, 3, 12, 7, 8, 6, 5, 20, 12, 14, 12, 10, 8, 33, 20, 24, 21, 20, 16, 13, 54, 33, 40, 36, 35, 32, 26, 21, 88, 54, 66, 60, 60, 56, 52, 42, 34, 143, 88, 108, 99, 100, 96, 91, 84, 68, 55, 232, 143, 176, 162, 165, 160, 156, 147, 136, 110, 89
Offset: 0

Views

Author

Gary W. Adamson, Mar 20 2005

Keywords

Examples

			The first few rows of the triangle are:
1;
2, 1;
4, 2, 2;
7, 4, 4, 3;
12, 7, 8, 6, 5;
20, 12, 14, 12, 10, 8

Crossrefs

Cf. A000071 (1st and 2nd column), A019274 (3rd column)

Formula

Matrix product of T(n,k) = sum_j A104762(n+1,j)*A104763(j+1,k), both interpreted as lower triangular square arrays.

Extensions

Incorrect conjecture on row sums removed. R. J. Mathar, Sep 17 2013

A104766 Triangle T(n,k) = A001629(n-k+2) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 10, 5, 2, 1, 20, 10, 5, 2, 1, 38, 20, 10, 5, 2, 1, 71, 38, 20, 10, 5, 2, 1, 130, 71, 38, 20, 10, 5, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 24 2005

Keywords

Comments

The triangle is the matrix square of the triangle A104762: T(n,k) = sum_{j= k..n} A104762(n,j)*A104762(j,k).

Examples

			First few rows of the triangle:
1;
2, 1;
5, 2, 1;
10, 5, 2, 1;
20, 10, 5, 2, 1;
38, 20, 10, 5, 2, 1;
71, 38, 20, 10, 5, 2, 1;
...

Crossrefs

Cf. A001629, A104762, A104763, A006478 (row sums).
Previous Showing 11-13 of 13 results.

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

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