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A004065 Define predecessors of n, P(n), to consist of numbers whose binary representation is obtained from that of n by replacing 10 with 01 or changing a final 1 to a 0; then a(0)=1, a(n) = Sum a(P(n)), n>0.

%I A004065 #28 Aug 28 2024 09:37:31
%S A004065 1,1,1,1,1,2,2,2,1,3,5,7,5,12,12,12,1,4,9,16,14,42,54,66,14,56,110,
%T A004065 176,110,286,286,286,1,5,14,30,28,100,154,220,42,198,462,858,572,1716,
%U A004065 2002,2288,42,240,702,1560,1274,4550,6552,8840,1274,5824,12376
%N A004065 Define predecessors of n, P(n), to consist of numbers whose binary representation is obtained from that of n by replacing 10 with 01 or changing a final 1 to a 0; then a(0)=1, a(n) = Sum a(P(n)), n>0.
%H A004065 Alois P. Heinz, <a href="/A004065/b004065.txt">Table of n, a(n) for n = 0..8191</a>
%F A004065 a(2^n-1) = A003121(n).
%e A004065 E.g. 201 = 11001001, so P(201) = {169, 197, 200}, a(201) = a(169) + a(197) + a(200).
%p A004065 P:= proc(n) local h, i, m, s, t;
%p A004065       t:= irem(n, 2, 'm');
%p A004065       s:= `if`(t=1, {n-1}, {});
%p A004065       for i from 0 while m>0 do h:= irem(m, 2, 'm');
%p A004065         if h=1 and t=0 then s:= s union {n- 2^i} fi; t:=h
%p A004065       od; s
%p A004065     end:
%p A004065 a:= proc(n) a(n):= `if`(n=0, 1, add(a(k), k=P(n))) end:
%p A004065 seq (a(n), n=0..80);  # _Alois P. Heinz_, Jul 06 2012
%t A004065 P[n_] := Module[{h, i, m, s, t}, {m, t} = QuotientRemainder[n, 2]; s = If[t == 1, {n-1}, {}]; For[i = 0, m>0, i++, {m, h} = QuotientRemainder[m, 2]; If[h == 1 && t == 0, s = s ~Union~ {n-2^i}]; t = h]; s]; a[n_] := a[n] = If[n == 0, 1, Sum[a[k], {k, P[n]}]]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jun 11 2015, after _Alois P. Heinz_ *)
%Y A004065 Cf. A003121.
%K A004065 nonn,base
%O A004065 0,6
%A A004065 _David W. Wilson_, Jan 29 2000
%E A004065 Entry revised by _N. J. A. Sloane_, Jun 14 2012