A073414 -id:A073414 - cp's OEIS frontend

A007400 Continued fraction for Sum_{n>=0} 1/2^(2^n) = 0.8164215090218931...

0, 1, 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 6, 4

Offset: 0

Simon Plouffe, Jeffrey Shallit

			0.816421509021893143708079737... = 0 + 1/(1 + 1/(4 + 1/(2 + 1/(4 + ...))))

M. Kmošek, Rozwinieçie Niektórych Liczb Niewymiernych na Ulamki Lancuchowe (Continued Fraction Expansion of Some Irrational Numbers), Master's thesis, Uniwersytet Warszawski, 1979.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Henry Cohn, Symmetry and specializability in continued fractions, Acta Arithmetica, volume 75, number 4, 1996, pages 297-320 (PDF). Also arXiv:math/0008221 [math.NT].
W. F. Lunnon, Q-D Tables and Zero-Squares, Manuscript, Jan. 1974. (Annotated scanned copy)
R. M. MacGregor, Generalizing the notion of a periodic sequence, American Math. Monthly 87 (1980), 90-102. (Annotated scanned copy)
B. Salvy, Email to N. J. A. Sloane, May 1994
Ratpoly info, May 1994
Jeffrey Shallit, Simple continued fractions for some irrational numbers, J. Number Theory 11 (1979), no. 2, 209-217.
Jeffrey O. Shallit, Simple continued fractions for some irrational numbers, J. Number Theory 11 (1979), no. 2, 209-217.
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A007404 (decimal), A073088 (partial sums), A073414/A073415 (convergents), A088431 (half), A089267, A092910.

a:= 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:
a(0):= 0: a(1):= 1: a(2):= 4:
map(a, [$0..1000]); # Robert Israel, Jun 14 2016

a[n_] := a[n] = Which[n < 3, {0, 1, 4}[[n+1]], Mod[n, 8] == 1, a[(n+1)/2], Mod[n, 8] == 2, a[(n+2)/2], True, {2, 0, 0, 2, 4, 4, 6, 4, 2, 0, 0, 2, 4, 6, 4, 4}[[Mod[n, 16]+1]]]; Table[a[n], {n, 0, 98}] (* Jean-François Alcover, Nov 29 2013, after Ralf Stephan *)

a(n)=if(n<3,[0,1,4][n+1],if(n%8==1,a((n+1)/2),if(n%8==2,a((n+2)/2),[2,0,0,2,4,4,6,4,2,0,0,2,4,6,4,4][(n%16)+1]))) /* Ralf Stephan */

a(n)=contfrac(suminf(n=0,1/2^(2^n)))[n+1]

{ allocatemem(932245000); default(realprecision, 26000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(x); for (n=1, 20001, write("b007400.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 07 2009

(define (A007400 n) (cond ((<= n 1) n) ((= 2 n) 4) (else (case (modulo n 8) ((0 3) 2) ((4 7) 4) ((1) (A007400 (/ (+ 1 n) 2))) ((2) (A007400 (/ (+ 2 n) 2))) (else (case (modulo n 16) ((5 14) 4) ((6 13) 6))))))) ;; (After Ralf Stephan's recurrence) - Antti Karttunen, Aug 12 2017

From Ralf Stephan, May 17 2005: (Start)
a(0)=0, a(1)=1, a(2)=4; for n > 2:
a(8k)    = a(8k+3)   = 2;
a(8k+4)  = a(8k+7)   = a(16k+5) = a(16k+14) = 4;
a(16k+6) = a(16k+13) = 6;
a(8k+1)  = a(4k+1);
a(8k+2)  = a(4k+2). (End)

A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

Original entry on oeis.org

1, 1, 5, 11, 49, 207, 1291, 5371, 12033, 53503, 333051, 719605, 3211471, 19988431, 83165195, 352649211, 788463617, 3506503679, 21827485691, 47161475061, 210473385935, 889055018801, 5544803498741, 23068269013765, 51681341526271

Offset: 1

Benoit Cloitre, Aug 23 2002

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

Cf. A007400, A007404, A073414.

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]:= 1: 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

(* 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] = 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 *)

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

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

cp's OEIS Frontend

A007400 Continued fraction for Sum_{n>=0} 1/2^(2^n) = 0.8164215090218931...

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A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

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Maple

Mathematica

PARI

Formula