A191488 A companion to Gould’s sequence A001316.

4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 66, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 68, 16, 24, 32, 40, 32, 48, 64, 72, 32, 48, 64, 80, 64, 96, 128

Offset

0,1

Comments

The row sums of the Sierpinski-Stern triangle A191372 are given by sequence A191487.

The differences diff1(n) = A191487(2*n+3) - A191487(2*n+1) lead to a peculiar number triangle, see the examples. The leading terms of the rows of the diff1(n) triangle clearly stand out from the rest of the terms and are given by A001550(p+1), p>=1; for p=0 this term is 7.

If we ignore the first term of the diff1(n) rows and reverse the order of the remaining terms we get sequence A191488, see the examples; more terms require a higher row number.

Both the diff1(n) and the diff2(n) sequences are related to Gould’s sequence A001316. We ignore the first term and reverse the order of the rest of the terms. The diff2(n) sequence leads directly to A001316, see A191487, while the diff1(n) sequence leads to A001316 in a slightly more complex way. We observe that for Gould’s sequence equation A001316((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^p holds, while for its companion A191488 equation A191488((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^(p+1)+2 holds; see the Maple program. Furthermore for both sequences a(2^p - 1) = C(p).

Links

Table of n, a(n) for n=0..62.

Formula

a((2*n+1)*2^p - 1) = C(p) * A001316(n) with C(p) = (2^(p+1)+2), p>=0.

a(2^p - 1) = 2^(p+1)+2 = A052548(p+1), p>=0.

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 diff1(n) reversed minus the first term are:
[4]
[4, 6, 8]
[4, 6, 8, 10, 8, 12, 16]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64]

Maple

nmax:=2^6; pmax:=ceil(log(nmax)/log(2)); A001316 := n -> if n<=-1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: C := proc(p): C(p) := 2^(p+1)+2 end: for p from 0 to pmax do for n from 0 to nmax do a((2*n+1)*2^p-1):= C(p)*A001316(n) od: od: seq(a(n), n=0..nmax-2);

Crossrefs

Cf. A001316, A191372, A191487.

Sequence in context: A277514 A331208 A362818 * A162600 A071537 A161396

Adjacent sequences: A191485 A191486 A191487 * A191489 A191490 A191491

Keyword

nonn,tabf

Author

Johannes W. Meijer, Jun 05 2011

Status

approved