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A137171 Interleaved reading of A000749 and its first to third differences.

%I A137171 #9 May 05 2024 17:30:08
%S A137171 0,0,0,1,0,0,1,1,0,1,2,1,1,3,3,1,4,6,4,2,10,10,6,6,20,16,12,16,36,28,
%T A137171 28,36,64,56,64,72,120,120,136,136,240,256,272,256,496,528,528,496,
%U A137171 1024,1056,1024,992,2080,2080,2016,2016,4160,4096,4032,4096,8256,8128,8128
%N A137171 Interleaved reading of A000749 and its first to third differences.
%C A137171 A000749 is identical to its fourth differences, which implies that the 2nd differences equal the 5th, the 3rd differences the 6th and so on and implies that each of the sequences of these differences obeys the recurrence a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3.
%C A137171 The table containing A000749 and its first differences (essentially A038505), 2nd differences (A038504) and 3rd differences (A038503) as the 4 rows is
%C A137171 O, 0, 0, 1, 4, 10, 20, 36, 64, ...
%C A137171 0, 0, 1, 3, 6, 10, 16, 28, 56, ...
%C A137171 0, 1, 2, 3, 4, 6, 12, 28, 64, ...
%C A137171 1, 1, 1, 1, 2, 6, 16, 36, 72, ...
%C A137171 Columns sums are 1, 2, 4, 8, 16, 32 ... = 2^n =A000079. The sequence reads this table column by column.
%H A137171 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4).
%t A137171 Join[{0, 0, 0, 1},LinearRecurrence[{0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4},{0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 1},59]] (* _Ray Chandler_, Sep 23 2015 *)
%Y A137171 Cf. A137172, A137173.
%K A137171 nonn,tabf
%O A137171 0,11
%A A137171 _Paul Curtz_, May 11 2008
%E A137171 Edited by _R. J. Mathar_, Jun 28 2008