A073414 Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

0, 1, 4, 9, 40, 169, 1054, 4385, 9824, 43681, 271910, 587501, 2621914, 16318985, 67897854, 287910401, 643718656, 2862785025, 17820428806, 38503642637, 171834999354, 725843640053, 4526896839672, 18833430998741, 42193758837154

Offset: 1

Author

Benoit Cloitre, Aug 23 2002

Keywords

• easy
• frac
• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..1650

Crossrefs

Cf. A007400, A007404, A073415.

Programs

Maple

a007400:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
A[1]:= 0: A[2]:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1]*a007400(n-1)+A[n-2];
od:
seq(A[n],n=1..100); # Robert Israel, Jun 14 2016

Mathematica

(* b is a007400 *)
b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
b[0] = 0; b[1] = 1; b[2] = 4;
a[1] = 0; a[2] = 1;
a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
Array[a, 100] (* Jean-François Alcover, Jun 10 2020, after Robert Israel *)

PARI

a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),1)

Formula

a(n) = a(n-1) A007400(n-1) + a(n-2). - Robert Israel, Jun 14 2016