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A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.

%I A317648 #25 Nov 01 2020 06:07:54
%S A317648 1,1,2,2,3,3,4,4,5,5,6,7,7,7,8,8,9,9,10,11,12,12,12,12,13,14,15,15,15,
%T A317648 15,16,16,17,17,18,19,20,21,21,21,21,21,22,23,24,25,26,27,27,27,27,27,
%U A317648 27,27,28,29,30,31,31,31,31,31,32,32,33,33,34,35,36,37,38,38,38,38,38,38,39,40,41,42,43,44,45
%N A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.
%C A317648 This sequence hits every positive integer.
%C A317648 Let b(1) = b(2) = b(3) = 1; for n >= 4, b(n) = b(t(n)) + b(n-t(n)) where t = A004001. Observe the symmetric relation between this sequence (a(n)) and b(n) thanks to line plots of a(n)-n/2 and b(n)-n/2 in Links section.
%H A317648 Altug Alkan, <a href="/A317648/b317648.txt">Table of n, a(n) for n = 1..65536</a>
%H A317648 Altug Alkan, <a href="/A317648/a317648_1.png">Line plot of a(n)-n/2 for n <= 2^17</a>
%H A317648 Altug Alkan, <a href="/A317648/a317648_2.png">Line plots of A004001(n)-n/2 and a(n)-n/2 for n <= 2^14</a>
%H A317648 Altug Alkan, <a href="/A317648/a317648_3.png">Line plots of a(n)-n/2 and b(n)-n/2 for n <= 2^11</a>
%F A317648 a(n+1) - a(n) = 0 or 1 for all n >= 1.
%p A317648 b:= proc(n) option remember; `if`(n<3, 1,
%p A317648       b(b(n-1)) +b(n-b(n-1)))
%p A317648     end:
%p A317648 a:= proc(n) option remember; `if`(n<3, 1,
%p A317648       a(b(n)) +a(n-b(n)))
%p A317648     end:
%p A317648 seq(a(n), n=1..100); # after _Alois P. Heinz_ at A317686
%t A317648 t[1] = 1; t[2] = 1; t[n_] := t[n] = t[t[n-1]] + t[n - t[n-1]];
%t A317648 a[1] = a[2] = 1; a[n_] := a[n] = a[t[n]] + a[n - t[n]];
%t A317648 Array[a, 100] (* _Jean-François Alcover_, Nov 01 2020 *)
%o A317648 (PARI) t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[t[n-1]]+t[n-t[n-1]]); a=vector(99); a[1]=a[2]=1; for(n=3, #a, a[n] = a[t[n]]+a[n-t[n]]); a
%Y A317648 Cf. A004001, A317686.
%K A317648 nonn,easy
%O A317648 1,3
%A A317648 _Altug Alkan_, Aug 02 2018