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A174980 Stern's diatomic series type ([0,1], 1). 4
0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 3, 1, 2, 1, 1, 0, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 5, 4, 7, 3, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A variant of Stern's diatomic series A002487. See the link [Luschny] and the Maple function below for the classification by types which is based on a generalization of Dijkstra's fusc function.

a(n) is also the number of superduperbinary integer partitions of n.

LINKS

Peter Luschny, row(n) for n = 0..12

Edsger Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. EWD 578: More about the function 'fusc'.

Peter Luschny, Rational Trees and Binary Partitions.

Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

FORMULA

Recursion: a(2n + 1) = a(n) and a(2n) = a(n - 1) + a(n) + [n = 2^k] for n = 1, a(0) = 0. [n = 2^k] is 1 if n is a power of 2, 0 otherwise.

EXAMPLE

The sequence splits into rows of length 2^k:

0,

0, 1,

0, 2, 1, 1,

0, 3, 2, 3, 1, 2, 1, 1,

0, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, ...

.

The first few partitions counted are:

[ 0], []

[ 1], []

[ 2], [[2]]

[ 3], []

[ 4], [[4], [2, 2]]

[ 5], [[4, 1]]

[ 6], [[4, 1, 1]]

[ 7], []

[ 8], [[8], [4, 4], [2, 2, 2, 2]]

[ 9], [[8, 1], [4, 4, 1]]

[10], [[8, 2], [8, 1, 1], [4, 4, 1, 1]]

[11], [[8, 2, 1]]

[12], [[8, 2, 2], [8, 2, 1, 1]]

[13], [[8, 2, 2, 1]]

[14], [[8, 2, 2, 1, 1]]

[15], []

[16], [[16], [8, 8], [4, 4, 4, 4], [2, 2, 2, 2, 2, 2, 2, 2]]

[17], [[16, 1], [8, 8, 1], [4, 4, 4, 4, 1]]

[18], [[16, 2], [8, 8, 2], [16, 1, 1], [8, 8, 1, 1], [4, 4, 4, 4, 1, 1]]

[19], [[16, 2, 1], [8, 8, 2, 1]]

[20], [[16, 4], [16, 2, 2], [8, 8, 2, 2], [16, 2, 1, 1], [8, 8, 2, 1, 1]]

[21], [[16, 4, 1], [16, 2, 2, 1], [8, 8, 2, 2, 1]]

[22], [[16, 4, 2], [16, 4, 1, 1], [16, 2, 2, 1, 1], [8, 8, 2, 2, 1, 1]]

[23], [[16, 4, 2, 1]]

[24], [[16, 4, 4], [16, 4, 2, 2], [16, 4, 2, 1, 1]]

MAPLE

SternDijkstra := proc(L, p, n) local k, i, len, M; len := nops(L); M := L; k := n; while k > 0 do M[1+(k mod len)] := add(M[i], i=1..len); k := iquo(k, len); od; op(p, M) end:

a := n -> SternDijkstra([0, 1], 1, n);

MATHEMATICA

a[0] = 0; a[n_?OddQ] := a[n] = a[(n-1)/2]; a[n_?EvenQ] := a[n] = a[n/2 - 1] + a[n/2] + Boole[ IntegerQ[ Log[2, n/2]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 26 2013 *)

PROG

(Sage)

def A174980(n):

    M = [0, 1]

    for b in n.bits():

        M[b] = M[0] + M[1]

    return M[0]

print([A174980(n) for n in (0..100)]) # Peter Luschny, Nov 28 2017

(Python3) # Generating the partitions.

def SDBinaryPartition(n):

    def Double(W, T):

        B = []

        for L in W:

            A = [a*2 for a in L]

            if T > 0: A += [1]*T

            B.append(A)

        return B

    if n == 2: return [[2]]

    if n <  4: return []

    h = n // 2

    H = SDBinaryPartition(h)

    B = Double(H, n % 2)

    if n % 2 == 0:

        H = SDBinaryPartition(h - 1)

        if H != []: B += Double(H, 2)

        if (n & (n - 1)) == 0: B.append([2]*h)

    return B

for n in range(25): print([n], SDBinaryPartition(n)) # Peter Luschny, Sep 02 2019

CROSSREFS

A002487, A070879, A047679, A007306, A174981, A140429 (row sums), A086449.

Sequence in context: A308067 A124748 A161225 * A277488 A325794 A119513

Adjacent sequences:  A174977 A174978 A174979 * A174981 A174982 A174983

KEYWORD

easy,nonn,tabf,look

AUTHOR

Peter Luschny, Apr 03 2010

STATUS

approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)