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A191487
The row sums of the Sierpinski-Stern triangle A191372.
3
0, 1, 3, 8, 9, 22, 24, 26, 27, 62, 66, 70, 72, 76, 78, 80, 81, 178, 186, 194, 198, 206, 210, 214, 216, 224, 228, 232, 234, 238, 240, 242, 243, 518, 534, 550, 558, 574, 582, 590, 594, 610, 618, 626, 630, 638, 642, 646
OFFSET
0,3
COMMENTS
The row sums a(n) of the Sierpinski-Stern triangle A191372 equal this sequence.
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.
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).
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.
FORMULA
a(2*n) = 3*a(n)
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
a(2^n+1) - a(2^n) = A085281(n+1) = A007689(n) for n>=0
a(2^(n+1)+1) - a(2^(n+1)-1) = A001550(n+1) for n>=1.
EXAMPLE
The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
[7]
[14, 4]
[36, 8, 6, 4]
[98, 16, 12, 8, 10, 8, 6, 4]
[276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
[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]
The first few rows of diff2(n) as a triangle, row lengths A011782(p) with p>=0, are:
[1]
[2]
[5, 1]
[13, 2, 2, 1]
[35, 4, 4, 2, 4, 2, 2, 1]
[97, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]
[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]
MAPLE
Add the following lines to the Maple program of A191372.
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);
KEYWORD
nonn
AUTHOR
Johannes W. Meijer, Jun 05 2011
STATUS
approved