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A322049 When A322050 is displayed as a triangle the rows converge to this sequence.

%I A322049 #37 Dec 16 2023 21:43:01
%S A322049 1,7,6,30,8,48,17,81,9,50,29,145,27,145,37,189,8,45,34,166,45,252,73,
%T A322049 342,37,179,89,425,74,374,86,412,8,49,33,165,46,270,91,436,50,277,149,
%U A322049 734,122,630,144,723,38,179,101,488,130,753,209,990,90,450,210,991
%N A322049 When A322050 is displayed as a triangle the rows converge to this sequence.
%C A322049 It would be nice to have a formula or recurrence. There is certainly a lot of structure.
%C A322049 Indices of records of a(n)/n are (1, 3, 7, 11, 23, 27, 43, 55, 87, 91, 119, 171, 183, 343, 347, 363, 367, 375, 439, 695, 731, 887, 1367, 1371, 1391, 1399, 1451, 1463, 2743, 2923, 2927, 2935, 3511, ...). The ratio a(n)/n increases roughly by 1 at each of these. We conjecture that this ratio is unbounded. We note that the record ratios occur in "clusters" at indices twice as large as the preceding cluster: 87, 91; 171, 183; 343..375; 695..731; 1367..1463; 2743..2935; ... This is compatible with the self-similar structure of the graph of this sequence, which starts over at a(2^k) = 8 for all k >= 4. (But note also the distinctive substructure repeating with period 2^10, cf. the "logarithmic plot" link.) - _M. F. Hasler_, Dec 18 2018
%H A322049 Hugo Pfoertner, <a href="/A322049/b322049.txt">Table of n, a(n) for n = 0..5461</a>
%H A322049 Hugo Pfoertner, <a href="/A322049/a322049.pdf">Logarithmic plot of 5462 terms</a>, use zoom to see details.
%F A322049 From _M. F. Hasler_, Dec 18 2018: (Start)
%F A322049 Experimental data suggests the following properties:
%F A322049 Sporadic values occurring only a finite number of times, with no regular pattern:
%F A322049   a(n) | 1 | 6 | 7 | 9 |   37   | 48 |  50   | 53 | ...
%F A322049   -----+---+---+---+---+--------+----+-------+----+-----
%F A322049     n  | 0 | 2 | 1 | 8 | 14, 24 |  5 | 9, 40 | 80 | ...
%F A322049 Values occurring in regular patterns:
%F A322049 a(n) = 8 iff n = 2^k, k = 2 or k >= 4; a(n) > 8 for all other n > 2.
%F A322049 a(n) = 33 iff n = 2^(2k+1) + 2, k >= 2; a(n) > 33 for all other n > 12 unless n = 2^k <=> a(n) = 8.
%F A322049 a(n) = 34 iff n = 4^k + 2, k >= 2.
%F A322049 a(n) = 38 iff n = 3*2^k, k = 4, 5, 6, 8, 10, ...
%F A322049 a(n) = 27*2^m if n = 3*2^k with k = 2 (m = 0) or k = 7, 9, ... (m = 1, 2, ...)
%F A322049 a(n) = 45 iff n = 20 or n = 4^k + 1, k >= 2.
%F A322049 a(n) = 46 iff n = 2^(2k+1) + 4, k >= 2.
%F A322049 a(n) = 49 iff n = 2^(2k+1) + 1, k >= 2, or n = 4^k + 4, k >= 3.
%F A322049 a(n) > 50 for all n > 10 not mentioned above. (End)
%Y A322049 Cf. A319018, A319019, A322048, A322050, A322051.
%K A322049 nonn,look
%O A322049 0,2
%A A322049 _N. J. A. Sloane_, Dec 15 2018