A174980 Stern's diatomic series type ([0,1], 1).

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

Offset: 0

Author

Peter Luschny, Apr 03 2010

Keywords

• easy
• nonn
• tabf
• look

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.

It appears that a(n) is equal to the multiplicative inverse of A002487(n+2) mod A002487(n+1). - Gary W. Adamson, Dec 23 2023

Examples

			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]]
		

Links

• Peter Luschny, row(n) for n = 0..12
• Edsger Dijkstra, EWD 578: More about the function 'fusc', Selected Writings on Computing, Springer, 1982, p. 232.
• Peter Luschny, Rational Trees and Binary Partitions.
• Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

Crossrefs

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

Programs

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 *)

Python

# 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

SageMath

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

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.