A022905 -id:A022905 - cp's OEIS frontend

A022907 The sequence m(n) in A022905.

0, 2, 5, 8, 14, 20, 29, 38, 53, 68, 89, 110, 140, 170, 209, 248, 302, 356, 425, 494, 584, 674, 785, 896, 1037, 1178, 1349, 1520, 1730, 1940, 2189, 2438, 2741, 3044, 3401, 3758, 4184, 4610, 5105, 5600, 6185, 6770, 7445, 8120, 8906, 9692, 10589

Offset: 1

Clark Kimberling

nonn

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.

a123[n_] := a123[n] = If[n == 0, 1, a123[Floor[n/2]] + a123[n-1]];
a[n_] := If[n == 1, 0, (3/2) a123[n-1] - 1]; Array[a, 50] (* Jean-François Alcover, Dec 04 2018 *)

from itertools import islice
from collections import deque
def A022907_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (0, 2, 5)
    while True:
        a += b
        yield 3*a-1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A022907_list = list(islice(A022907_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = 3 * A033485(n-1) - 1 = (3/2) * A000123(n-1) - 1, n>1. Proved by Jeremy Dover. - Ralf Stephan, Dec 08 2004

A022908 The sequence M(n) in A022905.

Original entry on oeis.org

0, 2, 5, 11, 20, 35, 56, 86, 125, 179, 248, 338, 449, 590, 761, 971, 1220, 1523, 1880, 2306, 2801, 3386, 4061, 4847, 5744, 6782, 7961, 9311, 10832, 12563, 14504, 16694, 19133, 21875, 24920, 28322, 32081, 36266, 40877, 45983, 51584

Offset: 1

Clark Kimberling

nonn

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.

(* b = A022905 *) b[1] = 1; b[n_] := b[n] = b[n-1] + 1 + If[EvenQ[n], 2 b[n/2], b[(n-1)/2] + b[(n+1)/2]];
a[1] = 0; a[n_] := b[n-1] + 1;
Array[a, 50] (* Jean-François Alcover, Nov 11 2018 *)

from itertools import islice
from collections import deque
def A022908_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (0,2)
    while True:
        a += b
        aqueue.append(a)
        if f:
            yield (3*a+1)//2
            b = aqueue.popleft()
        f = not f
A022908_list = list(islice(A022908_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = n + Sum_{k=1..n-1} A022907(k), n > 1. [corrected by Sean A. Irvine, May 22 2019]

a(1) = 0; a(n) = (1+3*A033485(2*n-3))/2 = A022905(n-1)+1, n > 1. - Philippe Deléham, May 30 2006

cp's OEIS Frontend

A022907 The sequence m(n) in A022905.

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