A022905 a(n) = M(n) + m(n) for n >= 2, where M(n) = max{ a(i) + a(n-i): i = 1..n-1 }, m(n) = min{ a(i) + a(n-i): i = 1..n-1 }.

1, 4, 10, 19, 34, 55, 85, 124, 178, 247, 337, 448, 589, 760, 970, 1219, 1522, 1879, 2305, 2800, 3385, 4060, 4846, 5743, 6781, 7960, 9310, 10831, 12562, 14503, 16693, 19132, 21874, 24919, 28321, 32080, 36265, 40876, 45982, 51583, 57769

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• J. M. Dover, On two OEIS conjectures, arXiv:1606.08033 [math.CO], 2016.

Crossrefs

Cf. A022907, A022908.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)+1+a(floor(n/2))+a(ceil(n/2)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 17 2013

Mathematica

a[n_] := a[n] = If[n==1, 1, a[n-1]+1+a[Floor[n/2]]+a[Ceiling[n/2]]]; Array[a,100] (* Jean-François Alcover, Aug 07 2017, after Alois P. Heinz *)

Python

from itertools import islice
from collections import deque
def A022905_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield 1
    while True:
        a += b
        aqueue.append(a)
        if f:
            yield (3*a-1)//2
            b = aqueue.popleft()
        f = not f
A022905_list = list(islice(A022905_gen(),40)) # Chai Wah Wu, Jun 08 2022

Formula

a(n) = n + Sum_{k=2..n} A022907(k).

a(n+1) = 1+3*Sum_{k=1..n} A033485(k). - Philippe Deléham, Jun 17 2010

a(n) = a(n-1) + 1 + a(floor(n/2)) + a(ceiling(n/2)) for n>1, a(1) = 1. - Alois P. Heinz, Sep 17 2013

a(n+1) = (3*A033485(2n+1)-1)/2. - Chai Wah Wu, Jun 08 2022