A376081 Irregular triangle read by rows: row n is the periodic part of the Leonardo numbers (A001595) modulo n.

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

Offset: 1

Author

Paolo Xausa, Sep 09 2024

Keywords

• nonn
• tabf

Comments

Each row n >= 3 ends in (1, n-1) (see Wikipedia article).

Examples

			Triangle begins:
  [1]  0;
  [2]  1;
  [3]  1, 1, 0, 2, 0, 0, 1, 2;
  [4]  1, 1, 3;
  [5]  1, 1, 3, 0, 4, 0, 0, 1, 2, 4, 2, 2, 0, 3, 4, 3, 3, 2, 1, 4;
  [6]  1, 1, 3, 5, 3, 3, 1, 5;
  [7]  1, 1, 3, 5, 2, 1, 4, 6, 4, 4, 2, 0, 3, 4, 1, 6;
  [8]  1, 1, 3, 5, 1, 7;
  [9]  1, 1, 3, 5, 0, 6, 7, 5, 4, 1, 6, 8, 6, 6, 4, 2, 7, 1, 0, 2, 3, 6, 1, 8;
  ...
For n = 8:
A001595       = 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, ...
A001595 mod 8 = 1, 1, 3, 5, 1,  7,  1,  1,  3,   5,   1,   7,   1, ...
                \_______________/
                  periodic part
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12347 (rows 1..150 of triangle, flattened).
• Wikipedia, Leonardo number.

Crossrefs

Cf. A001595, A161553, A376082 (row lengths), A376083 (row sums).

Programs

Mathematica

A376081row[n_] := If[n < 3, {n - 1}, Module[{k = 1}, NestWhileList[Mod[2 * Fibonacci[++k] - 1, n] &, 1, {#, #2} != {1, n-1} &, {3, 2}]]];
Array[A376081row, 10]

Formula

T(n,k) = A001595(k) mod n, with 0 <= k < A376082(n).