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A104763
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Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.
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14
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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
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OFFSET
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1,6
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COMMENTS
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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
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LINKS
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Reinhard Zumkeller, Rows n = 1..100 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
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FORMULA
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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
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EXAMPLE
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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; ...
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MATHEMATICA
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Table[Fibonacci[k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Gary W. Adamson, Mar 23 2005
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EXTENSIONS
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Edited by R. J. Mathar, May 02 2008
Extended by R. J. Mathar, Aug 27 2008
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STATUS
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approved
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