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A191487 The row sums of the Sierpinski-Stern triangle A191372.

%I A191487 #6 Mar 30 2012 18:59:51
%S A191487 0,1,3,8,9,22,24,26,27,62,66,70,72,76,78,80,81,178,186,194,198,206,
%T A191487 210,214,216,224,228,232,234,238,240,242,243,518,534,550,558,574,582,
%U A191487 590,594,610,618,626,630,638,642,646
%N A191487 The row sums of the Sierpinski-Stern triangle A191372.
%C A191487 The row sums a(n) of the Sierpinski-Stern triangle A191372 equal this sequence.
%C A191487 The differences diff1(n) = a(2*n+3) - a(2*n+1) and diff2(n) = (a(2*n+2) - a(2*n))/3, give rise to patterns that lead to Gould’s sequence A001316, see the examples.
%C A191487 The diff1(n) sequence as a triangle leads to Gould’s sequence in a peculiar way, see A191488. The leading terms of the diff1(n) rows are given by A001550(p+1), p>=1; for p=0 the leading term is 7. The rows sums of diff1(n) as a triangle equal A025192(p+2), p>=1; for p = 0 the row sum is 7. The row sums of diff1(n) as a triangle minus the first term equal 2*A053152(p+1).
%C A191487 The diff2(n) sequence as a triangle leads to Gould’s sequence A001316 in a simple way; just delete the first term and reverse the order of the rest of the terms; more terms require a higher row number. The leading terms of the diff2(n) rows are given by A085281(p), p>=0. The row sums of diff2(n) as a triangle equal A025192(p) and the row sums minus the first term equal A001047(p-1), p>=1; for p=0 the row sum minus the first term is 0.
%F A191487 a(2*n) = 3*a(n)
%F A191487 diff(n) = a(n+1) - a(n), diff1(n) = a(2*n+3) - a(2*n+1), diff2(n) = (a(2*n+2) - a(2*n))/3
%F A191487 a(2^n+1) - a(2^n) = A085281(n+1) = A007689(n) for n>=0
%F A191487 a(2^(n+1)+1) - a(2^(n+1)-1) = A001550(n+1) for n>=1.
%e A191487 The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
%e A191487 [7]
%e A191487 [14, 4]
%e A191487 [36, 8, 6, 4]
%e A191487 [98, 16, 12, 8, 10, 8, 6, 4]
%e A191487 [276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
%e A191487 [794, 64, 48, 32, 40, 32, 24, 16, 36, 32, 24, 16, 20, 16, 12, 8, 34, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
%e A191487 The first few rows of diff2(n) as a triangle, row lengths A011782(p) with p>=0, are:
%e A191487 [1]
%e A191487 [2]
%e A191487 [5, 1]
%e A191487 [13, 2, 2, 1]
%e A191487 [35, 4, 4, 2, 4, 2, 2, 1]
%e A191487 [97, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]
%e A191487 [275, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4, 4, 2, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]
%p A191487 Add the following lines to the Maple program of A191372.
%p A191487 A191487(0):=0: for d from 1 to 2^pmax do A191487(d):= 0: for Tx from 0 to 2^ceil(log(d)/ log(2))-1 do A191487(d):=A191487(d)+S2(Tx,d) od: od: seq(A191487(d),d=0..2^pmax);
%Y A191487 Cf. A191372, A191488, A001316, A001550, A025192, A053152, A085281, A001047, A007689.
%K A191487 nonn
%O A191487 0,3
%A A191487 _Johannes W. Meijer_, Jun 05 2011