A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

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

Offset

1,6

Comments

Triangle of A104762, Fibonacci sequence in each row starts from the right.

The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - Johannes W. Meijer, Sep 22 2010

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

Links

Reinhard Zumkeller, Rows n = 1..100 of table, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

F(1) through F(n) starting from the left in n-th row.

T(n,k) = A000045(k), 1<=k<=n. - R. J. Mathar, May 02 2008

a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

Example

First few rows of the triangle are:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 5;
  1, 1, 2, 3, 5, 8;
  1, 1, 2, 3, 5, 8, 13; ...

Mathematica

Table[Fibonacci[k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)

Prog

(Haskell)
a104763 n k = a104763_tabl !! (n-1) !! (k-1)
a104763_row n = a104763_tabl !! (n-1)
a104763_tabl = map (flip take $ tail a000045_list) [1..]
-- Reinhard Zumkeller, Aug 15 2013
(PARI) for(n=1, 15, for(k=1, n, print1(fibonacci(k), ", "))) \\ G. C. Greubel, Jul 13 2019
(Magma) [Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019
(Sage) [[fibonacci(k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019
(GAP) Flat(List([1..15], n-> List([1..n], Fibonacci(k) ))) # G. C. Greubel, Jul 13 2019

Crossrefs

Cf. A104762, A000045.

Cf. A000071 (row sums). - R. J. Mathar, Jul 22 2009

Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - Johannes W. Meijer, Sep 22 2010

Sequence in context: A324209 A228107 A140207 * A027751 A181322 A004070

Adjacent sequences: A104760 A104761 A104762 * A104764 A104765 A104766

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Mar 23 2005

Extensions

Edited by R. J. Mathar, May 02 2008

Extended by R. J. Mathar, Aug 27 2008

Status

approved