cp's OEIS Frontend

A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.

%I A351123 #74 Jul 18 2022 09:49:59
%S A351123 1,5,4,1,2,4,7,11,2,3,4,6,9,13,1,3,6,10,3,7,1,2,3,8,12,2,5,9,1,4,5,7,
%T A351123 10,14,6,1,2,7,11,2,3,6,7,9,12,16,1,3,4,5,6,7,9,11,12,14,15,16,18,19,
%U A351123 20,21,23,26,27,28,30,31,33,34,35,36,37,38,41,42,43,44,48,50,52,56,59,60,61,66,70
%N A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.
%C A351123 The terms in row n are T(n,0), T(n,1), ..., T(n, A258145(n)-2), and are the partial sums of the terms in row n of A351122.
%C A351123 In each row n, the terms also satisfy the equation 3* (3* (3* (3* ... (3* (2n+1) +1) + 2^T(n,0)) + 2^T(n,1)) + 2^T(n,2)) + ... = 2^T(n, A258145(n)-2); e.g., for n=4, and A258145(4)-2=5: 3* (3* (3* (3* (3* (3*9+1) +2^2) +2^3) +2^4) +2^6) +2^9 = 2^13.
%C A351123 For row n, the right-hand side of the equation above is 2^A166549(n+1). E.g., for the above example (n=4), the right-hand side is 2^A166549(4+1) = 2^13.
%e A351123 Triangle starts at T(1,0):
%e A351123 n\k   0   1   2   3   4   5   6   7   8   9   10 ...
%e A351123 1:    1   5
%e A351123 2:    4
%e A351123 3:    1   2   4   7  11
%e A351123 4:    2   3   4   6   9  13
%e A351123 5:    1   3   6  10
%e A351123 6:    3   7
%e A351123 7:    1   2   3   8   12
%e A351123 8:    2   5   9
%e A351123 ...
%e A351123 E.g., row 3 of A351122 is [1, 1, 2, 3, 4]; its partial sums are [1, 2, 4, 7, 11].
%o A351123 (PARI) orow(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ A351122
%o A351123 row(n) = my(v = orow(n)); vector(#v, k, sum(i=1, k, v[i])); \\ _Michel Marcus_, Jul 18 2022
%Y A351123 Cf. A351122, A256598, A258145, A087230, A166549, A075680.
%K A351123 nonn,tabf
%O A351123 1,2
%A A351123 _Flávio V. Fernandes_, Feb 01 2022
%E A351123 Data corrected by _Mohsen Maesumi_, Jul 18 2022
%E A351123 Last row completed by _Michel Marcus_, Jul 18 2022