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%I #33 Sep 08 2022 08:45:17
%S 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,
%T 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,
%U 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
%N Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.
%C Triangle of A104762, Fibonacci sequence in each row starts from the right.
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A104763/b104763.txt">Rows n = 1..100 of table, flattened</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F F(1) through F(n) starting from the left in n-th row.
%F T(n,k) = A000045(k), 1<=k<=n. - _R. J. Mathar_, May 02 2008
%F a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 13 2012
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 2, 3;
%e 1, 1, 2, 3, 5;
%e 1, 1, 2, 3, 5, 8;
%e 1, 1, 2, 3, 5, 8, 13; ...
%t Table[Fibonacci[k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 13 2019 *)
%o (Haskell)
%o a104763 n k = a104763_tabl !! (n-1) !! (k-1)
%o a104763_row n = a104763_tabl !! (n-1)
%o a104763_tabl = map (flip take $ tail a000045_list) [1..]
%o -- _Reinhard Zumkeller_, Aug 15 2013
%o (PARI) for(n=1,15, for(k=1,n, print1(fibonacci(k), ", "))) \\ _G. C. Greubel_, Jul 13 2019
%o (Magma) [Fibonacci(k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 13 2019
%o (Sage) [[fibonacci(k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Jul 13 2019
%o (GAP) Flat(List([1..15], n-> List([1..n], Fibonacci(k) ))) # _G. C. Greubel_, Jul 13 2019
%Y Cf. A104762, A000045.
%Y Cf. A000071 (row sums). - _R. J. Mathar_, Jul 22 2009
%Y 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
%K nonn,tabl,easy
%O 1,6
%A _Gary W. Adamson_, Mar 23 2005
%E Edited by _R. J. Mathar_, May 02 2008
%E Extended by _R. J. Mathar_, Aug 27 2008
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