%I #52 Nov 05 2023 09:04:17
%S 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,
%T 26,26,26,26,26,31,31,33,34,35,35,35,35,39,40,40,40,43,44,44,44,47,47,
%U 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
%N a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.
%C Critical values of Conway's game of one-dimensional phutball (A037988), except for the initial zero.
%C Conjectured run lengths are A272729, and the terms which occur here are partial sums of A272729.
%C The next distinct term here occurs at index a(i)+i+1 for every index i.
%H Neal Gersh Tolunsky, <a href="/A359807/b359807.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%e 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.
%o (PARI) lista(nn) = my(va = vector(nn)); va[1] = 0; for (n=2, nn, va[n] = vecmax(select(x->(x<n), vector(n-1, i, va[i]+i)));); va; \\ _Michel Marcus_, Jan 31 2023
%o (PARI) { 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
%Y Cf. A367039, A037988, A272729, A272727, A094591, A367026.
%K nonn
%O 1,4
%A _Neal Gersh Tolunsky_, Jan 13 2023