A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

1, 3, 5, 9, 13, 19, 25, 35, 45, 59, 73, 93, 113, 139, 165, 201, 237, 283, 329, 389, 449, 523, 597, 691, 785, 899, 1013, 1153, 1293, 1459, 1625, 1827, 2029, 2267, 2505, 2789, 3073, 3403, 3733, 4123, 4513, 4963, 5413, 5937, 6461, 7059, 7657, 8349

Offset: 1

Author

Mitch Harris, Jan 05 2005

Keywords

• nonn

Comments

From Gus Wiseman, Mar 23 2019: (Start)

The offset could safely be changed to zero by setting the boundary condition to a(0) = 0.

Also the number of integer partitions of 2n into powers of 2 with at least one part > 1. The Heinz numbers of these partitions are given by A324927. For example, the a(1) = 1 through a(5) = 13 integer partitions are:

(2) (4) (42) (8) (82)

(22) (222) (44) (442)

(211) (411) (422) (811)

(2211) (2222) (4222)

(21111) (4211) (4411)

(22211) (22222)

(41111) (42211)

(221111) (222211)

(2111111) (421111)

(2221111)

(4111111)

(22111111)

(211111111)

(End)

Crossrefs

Cf. A000123, A033485.

Cf. A018819, A318400, A324927, A324928.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And[Max@@#>1,And@@IntegerQ/@Log[2,#]]&]],{n,0,30,2}] (* Gus Wiseman, Mar 23 2019 *)

Python

from itertools import islice
from collections import deque
def A102378_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3)
    while True:
        a += b
        yield 2*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A102378_list = list(islice(A102378_gen(),40)) # Chai Wah Wu, Jun 08 2022

Formula

a(n) - a(n-1) = A018819(n+1)

G.f. A(x) satisfies (1-x)*A(x) = 2(1 + x)*B(x^2), where B(x) is the gf of A033485

a(n) = A000123(n) - 1. - Gus Wiseman, Mar 23 2019

G.f. A(x) satisfies: A(x) = (x + (1 - x^2) * A(x^2)) / (1 - x)^2. - Ilya Gutkovskiy, Aug 11 2021