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A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

%I A381962 #11 Mar 14 2025 21:19:40
%S A381962 0,1,2,1,3,2,1,4,1,5,2,1,6,2,1,7,3,2,1,8,1,9,2,1,10,2,1,11,3,2,1,12,2,
%T A381962 1,13,3,2,1,14,3,2,1,15,4,1,16,1,17,2,1,18,2,1,19,3,2,1,20,2,1,21,3,2,
%U A381962 1,22,3,2,1,23,4,1,24,2,1,25,3,2,1,26,3,2,1
%N A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).
%H A381962 Paolo Xausa, <a href="/A381962/b381962.txt">Table of n, a(n) for n = 0..11724</a> (rows 0..3000 of triangle, flattened).
%F A381962 T(n,0) = n and, for k = 1..A180094(n), T(n,k) = A000120(T(n,k-1)).
%e A381962 Triangle begins:
%e A381962   n\k|  0  1  2  3
%e A381962   ----------------
%e A381962    0 |  0;
%e A381962    1 |  1;
%e A381962    2 |  2, 1;
%e A381962    3 |  3, 2, 1;
%e A381962    4 |  4, 1;
%e A381962    5 |  5, 2, 1;
%e A381962    6 |  6, 2, 1;
%e A381962    7 |  7, 3, 2, 1;
%e A381962    8 |  8, 1;
%e A381962    9 |  9, 2, 1;
%e A381962   10 | 10, 2, 1;
%e A381962   ...
%t A381962 A381962row[n_] := NestWhileList[DigitSum[#, 2] &, n, # > 1 &];
%t A381962 Array[A381962row, 30, 0]
%o A381962 (Python)
%o A381962 def row(n):
%o A381962     out = [n] if n > 1 else []
%o A381962     while (n:=n.bit_count()) > 1:
%o A381962         out += [n]
%o A381962     return out + [n]
%o A381962 print([e for n in range(27) for e in row(n)]) # _Michael S. Branicky_, Mar 12 2025
%Y A381962 Cf. A000120, A078627 (row lengths), A078677 (row sums), A180094.
%Y A381962 Cf. A381963, A381965.
%K A381962 nonn,tabf,base,easy
%O A381962 0,3
%A A381962 _Paolo Xausa_, Mar 11 2025