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

%I A376081 #21 Sep 13 2024 08:11:25
%S A376081 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,
%T A376081 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,
%U A376081 6,7,5,4,1,6,8,6,6,4,2,7,1,0,2,3,6,1,8
%N A376081 Irregular triangle read by rows: row n is the periodic part of the Leonardo numbers (A001595) modulo n.
%C A376081 Each row n >= 3 ends in (1, n-1) (see Wikipedia article).
%H A376081 Paolo Xausa, <a href="/A376081/b376081.txt">Table of n, a(n) for n = 1..12347</a> (rows 1..150 of triangle, flattened).
%H A376081 Wikipedia, <a href="https://en.wikipedia.org/wiki/Leonardo_number">Leonardo number</a>.
%F A376081 T(n,k) = A001595(k) mod n, with 0 <= k < A376082(n).
%e A376081 Triangle begins:
%e A376081   [1]  0;
%e A376081   [2]  1;
%e A376081   [3]  1, 1, 0, 2, 0, 0, 1, 2;
%e A376081   [4]  1, 1, 3;
%e A376081   [5]  1, 1, 3, 0, 4, 0, 0, 1, 2, 4, 2, 2, 0, 3, 4, 3, 3, 2, 1, 4;
%e A376081   [6]  1, 1, 3, 5, 3, 3, 1, 5;
%e A376081   [7]  1, 1, 3, 5, 2, 1, 4, 6, 4, 4, 2, 0, 3, 4, 1, 6;
%e A376081   [8]  1, 1, 3, 5, 1, 7;
%e A376081   [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;
%e A376081   ...
%e A376081 For n = 8:
%e A376081 A001595       = 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, ...
%e A376081 A001595 mod 8 = 1, 1, 3, 5, 1,  7,  1,  1,  3,   5,   1,   7,   1, ...
%e A376081                 \_______________/
%e A376081                   periodic part
%t A376081 A376081row[n_] := If[n < 3, {n - 1}, Module[{k = 1}, NestWhileList[Mod[2 * Fibonacci[++k] - 1, n] &, 1, {#, #2} != {1, n-1} &, {3, 2}]]];
%t A376081 Array[A376081row, 10]
%Y A376081 Cf. A001595, A161553, A376082 (row lengths), A376083 (row sums).
%K A376081 nonn,tabf
%O A376081 1,6
%A A376081 _Paolo Xausa_, Sep 09 2024