A367026 -id:A367026 - cp's OEIS frontend

A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

0, 1, 1, 3, 4, 4, 4, 7, 7, 9, 10, 11, 11, 11, 11, 15, 16, 16, 16, 19, 19, 21, 21, 23, 24, 25, 26, 26, 26, 26, 26, 31, 31, 33, 34, 35, 35, 35, 35, 39, 40, 40, 40, 43, 44, 44, 44, 47, 47, 49, 49, 51, 51, 53, 54, 55, 56, 57, 57, 57, 57, 57, 57, 63, 64, 64, 64, 67, 67, 69, 69, 71, 72, 73, 74, 74, 74, 74, 74

Offset: 1

Neal Gersh Tolunsky, Jan 13 2023

nonn

Critical values of Conway's game of one-dimensional phutball (A037988), except for the initial zero.
Conjectured run lengths are A272729, and the terms which occur here are partial sums of A272729.
The next distinct term here occurs at index a(i)+i+1 for every index i.

			For n=3, we see that for i=1 and 2, a(i)+i = 1 and 3, of which only 1 is < n=3, so that a(3)=1.
For n=5, i=1..4 have a(i)+i = 1,3,4,7 and the largest < n=5 is 4 so that a(5)=4.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A367039, A037988, A272729, A272727, A094591, A367026.

lista(nn) = my(va = vector(nn)); va[1] = 0; for (n=2, nn, va[n] = vecmax(select(x->(xMichel Marcus, Jan 31 2023

{ my (v = 0, m = 0); for (n = 1, 79, if (bittest(m, n-1), v = n-1); print1 (v", "); m = bitor(m, 2^(v+n))) } \\ Rémy Sigrist, Feb 08 2023

A367039 a(1) = 0, a(2) = 1; thereafter a(n) is the largest index < n not equal to i +- a(i) for any i = 1..n-1.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 7, 8, 8, 8, 8, 12, 13, 14, 14, 16, 16, 16, 16, 16, 21, 22, 23, 24, 24, 26, 26, 28, 28, 28, 31, 32, 32, 32, 32, 32, 32, 38, 39, 40, 41, 42, 42, 44, 44, 46, 46, 48, 48, 48, 51, 52, 52, 52, 55, 56, 56, 56, 56, 60, 61, 62, 62, 64, 64, 64, 64, 64, 64, 64

Offset: 1

Neal Gersh Tolunsky, Nov 02 2023

nonn

It appears that A085262 gives the distinct values of this sequence (except for the initial zero).

The sequence is nondecreasing.

			a(8)=7 because 7 is the largest index that cannot be expressed as the sum a(i)+-i for any i < n. 4 also cannot be expressed in this way, but it is smaller than 7.
Another way to see this is to consider each index i as a location from which one can jump a(i) terms forward or backward. For a(8)=7, we find the largest index which cannot be reached in this way (a smaller index being i=4):
0, 1, 2, 2, 4, 4, 4
0<-1
0, 1, 2, 2, 4, 4, 4
   1<----2
0, 1, 2, 2, 4, 4, 4
   1->2<----------4
0, 1, 2, 2, 4, 4, 4
         ?
0, 1, 2, 2, 4, 4, 4
      2---->4
0, 1, 2, 2, 4, 4, 4
         2---->4
0, 1, 2, 2, 4, 4, 4
                  ?

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A359807, A085262, A367026.

cp's OEIS Frontend

A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

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