A317686 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A063882.

1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 11, 12, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 25, 26, 27, 27, 27, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43, 44, 45, 46, 46, 47, 48, 49, 49, 49, 49, 50, 51

Offset: 1

Author

Altug Alkan, Aug 04 2018

Keywords

• nonn

Comments

This sequence hits every positive integer and it has a fractal-like structure, see scatterplot of 2*n-3*a(n) in Links section.

Let b(1) = b(2) = b(3) = b(4) = 1; for n >= 5, b(n) = b(t(n)) + b(n-t(n)) where t = A063882. Observe the symmetric relation between this sequence (a(n)) and b(n) thanks to plots of a(n)-2*n/3 and b(n)-n/3 in Links section. Note that a(n) + b(n) = n for n >= 2.

Links

• Altug Alkan, Scatterplot of 2*n-3*a(n) for n <= 36000
• Altug Alkan, Scatterplots of a(n)-2*n/3 and b(n)-n/3 for n <= 36000

Crossrefs

Cf. A063882, A317648.

Programs

Maple

b:= proc(n) option remember; `if`(n<5, 1,
      b(n-b(n-1)) +b(n-b(n-4)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      a(b(n)) +a(n-b(n)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 05 2018

Mathematica

b[n_] := b[n] = If[n < 5, 1, b[n - b[n - 1]] + b[n - b[n - 4]]];
a[n_] := a[n] = If[n < 3, 1, a[b[n]] + a[n - b[n]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

PARI

t=vector(99); t[1]=t[2]=t[3]=t[4]=1; for(n=5, #t, t[n] = t[n-t[n-1]]+t[n-t[n-4]]); 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.