A160271 -id:A160271 - cp's OEIS frontend

A199474 Leftmost column in the monotonic justified array of all positive generalized Fibonacci sequences (A160271).

1, 2, 3, 2, 4, 3, 5, 4, 3, 6, 5, 4, 7, 6, 5, 4, 8, 7, 6, 5, 9, 8, 7, 6, 5, 10, 9, 8, 7, 6, 11, 10, 9, 8, 7, 6, 12, 11, 10, 9, 8, 7, 13, 12, 11, 10, 9, 8, 7, 14, 13, 12, 11, 10, 9, 8, 15, 14, 13, 12, 11, 10, 9, 8, 16, 15, 14, 13, 12, 11, 10, 9, 17, 16, 15, 14

Offset: 1

Casey Mongoven, Nov 06 2011

nonn

Northwest corner of A160271:
  1,   0,   1,   1,   2,   3,   5,   8,  13,  21, ...
  2,   0,   2,   2,   4,   6,  10,  16,  26,  42, ...
  3,   0,   3,   3,   6,   9,  15,  24,  39,  63, ...
  2,   1,   3,   4,   7,  11,  18,  29,  47,  76, ...
  4,   0,   4,   4,   8,  12,  20,  32,  52,  84, ...
  3,   1,   4,   5,   9,  14,  23,  37,  60,  97, ...
  5,   0,   5,   5,  10,  15,  25,  40,  65, 105, ...
  4,   1,   5,   6,  11,  17,  28,  45,  73, 118, ...
  3,   2,   5,   7,  12,  19,  31,  50,  81, 131, ...
  ...

Clark Kimberling, Orderings of the set of all positive Fibonacci sequences, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.

Cf. A160271, A199087, A199088, A339399.

a(n) = (s(n)^2 - n) * [s(n)^2 - s(n) >= n] + (s(n)^2 - n + s(n)) * [s(n)^2 - s(n) < n] where s(n) = ceiling(sqrt(n)). - Iliya Trub, Mar 17 2019
a(n) = A339399(2n). - Wesley Ivan Hurt, Jan 09 2022
a(n) = floor(ceiling(sqrt(4n))^2/4)+floor(sqrt(4n-2))-floor(sqrt(n)+1/2)-n+1. - Wesley Ivan Hurt, Jan 09 2022

A199087 The smallest row index of a row in A160271 (the monotonic justified array of generalized Fibonacci sequences) which contains n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 4, 1, 3, 2, 4, 5, 1, 6, 3, 2, 8, 4, 9, 5, 1, 12, 6, 3, 7, 2, 16, 8, 4, 10, 9, 5, 11, 1, 13, 12, 6, 14, 3, 7, 15, 2, 18, 16, 8, 19, 4, 10, 20, 9, 23, 5, 11, 24, 1, 13, 25, 12, 29, 6, 14, 30, 3, 17, 7, 15, 35, 2, 18, 36, 16, 21, 8, 19, 42

Offset: 0

Casey Mongoven, Nov 06 2011

nonn

Northwest corner of A160271, including the Fibonacci-type sequence containing only 0's as the 0th row:
...0....0....0....0....0....0....0....0....0
...0....1....1....2....3....5....8...13...21
...0....2....2....4....6...10...16...26...42
...0....3....3....6....9...15...24...39...63
...1....3....4....7...11...18...29...47...76
...0....4....4....8...12...20...32...52...84
...1....4....5....9...14...23...37...60...97
...0....5....5...10...15...25...40...65..105
...1....5....6...11...17...28...45...73..118
...2....5....7...12...19...31...50...81..131

Clark Kimberling, Orderings of the set of all positive Fibonacci sequences, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.

Cf. A160271, A019586, A035513.

A199088 Excluding the two leftmost columns - the seeds - in the monotonic justified array of all positive Fibonacci sequences (A160271), the lowest column index which contains n in the smallest row which contains n (A199087).

Original entry on oeis.org

2, 2, 4, 5, 4, 6, 5, 4, 7, 5, 6, 5, 5, 8, 5, 6, 7, 5, 6, 5, 6, 9, 5, 6, 7, 6, 8, 5, 6, 7, 6, 6, 7, 6, 10, 6, 6, 7, 6, 8, 7, 6, 9, 6, 6, 7, 6, 8, 7, 6, 7, 6, 8, 7, 6, 11, 7, 6, 7, 6, 8, 7, 6, 9, 7, 8, 7, 6, 10, 7, 6, 7, 7, 8, 7, 6, 9, 7, 8, 7, 7, 8, 7, 6, 9

Offset: 0

Casey Mongoven, Nov 06 2011

nonn

Northwest corner of A160271, including the Fibonacci-type sequence containing only 0's as the 0th row:
...0....0....0....0....0....0....0....0....0
...0....1....1....2....3....5....8...13...21
...0....2....2....4....6...10...16...26...42
...0....3....3....6....9...15...24...39...63
...1....3....4....7...11...18...29...47...76
...0....4....4....8...12...20...32...52...84
...1....4....5....9...14...23...37...60...97
...0....5....5...10...15...25...40...65..105
...1....5....6...11...17...28...45...73..118
...2....5....7...12...19...31...50...81..131

Clark Kimberling, Orderings of the set of all positive Fibonacci sequences, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.

Cf. A199087, A160271, A035614.

A055087 Integers 0..n then 0..n then 0..n+1 then 0..n+1 etc.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1

Offset: 0

Michael Somos, Jun 13 2000

Second column in the monotonic justified array of all positive generalized Fibonacci sequences (A160271). see also A199474. - Casey Mongoven, Nov 06 2011

Can also be seen as an irregular triangle with duplicated rows of A002262. - Reinhard Zumkeller, Jul 14 2015

			0; 0; 0,1; 0,1; 0,1,2; 0,1,2; 0,1,2,3; 0,1,2,3; ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002262.

import Data.List (transpose)
a055087 n k = a055087_tabf !! n !! k
a055087_row n = a055087_tabf !! n
a055087_tabf = concat $ transpose [a002262_tabl, a002262_tabl]
a055087_list = concat a055087_tabf
-- Reinhard Zumkeller, Jul 14 2015

With[{nn=10},Flatten[Table[{Range[0,n],Range[0,n]},{n,0,nn}]]] (* Harvey P. Dale, May 30 2015 *)

{a(n) = (1 + 4*n - sqr( floor( sqrt( 1 + 4*n)))) \ 4}

cp's OEIS Frontend

A199474 Leftmost column in the monotonic justified array of all positive generalized Fibonacci sequences (A160271).

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A199087 The smallest row index of a row in A160271 (the monotonic justified array of generalized Fibonacci sequences) which contains n.

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A199088 Excluding the two leftmost columns - the seeds - in the monotonic justified array of all positive Fibonacci sequences (A160271), the lowest column index which contains n in the smallest row which contains n (A199087).

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A055087 Integers 0..n then 0..n then 0..n+1 then 0..n+1 etc.

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