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

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, 15, 16, 16, 17, 17, 18, 19, 20, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 27, 27, 27, 27, 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

Offset: 1

Author

Altug Alkan, Aug 02 2018

Keywords

• nonn
• easy

Comments

This sequence hits every positive integer.

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.

Links

• Altug Alkan, Table of n, a(n) for n = 1..65536
• Altug Alkan, Line plot of a(n)-n/2 for n <= 2^17
• Altug Alkan, Line plots of A004001(n)-n/2 and a(n)-n/2 for n <= 2^14
• Altug Alkan, Line plots of a(n)-n/2 and b(n)-n/2 for n <= 2^11

Crossrefs

Cf. A004001, A317686.

Programs

Maple

b:= proc(n) option remember; `if`(n<3, 1,
      b(b(n-1)) +b(n-b(n-1)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      a(b(n)) +a(n-b(n)))
    end:
seq(a(n), n=1..100); # after Alois P. Heinz at A317686

Mathematica

t[1] = 1; t[2] = 1; t[n_] := t[n] = t[t[n-1]] + t[n - t[n-1]];
a[1] = a[2] = 1; a[n_] := a[n] = a[t[n]] + a[n - t[n]];
Array[a, 100] (* Jean-François Alcover, Nov 01 2020 *)

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

Formula

a(n+1) - a(n) = 0 or 1 for all n >= 1.