A051135 a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001.

2, 2, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3

Offset: 1

Author

Robert Lozyniak (11(AT)onna.com)

Keywords

• easy
• nonn
• nice

Comments

If the initial 2 is changed to a 1, the resulting sequence (A265332) has the property that if all 1's are deleted, the remaining terms are the sequence incremented. - Franklin T. Adams-Watters, Oct 05 2006

a(A088359(n)) = 1 and a(A087686(n)) > 1; first differences of A188163. - Reinhard Zumkeller, Jun 03 2011

From Robert G. Wilson v, Jun 07 2011: (Start)

a(k)=1 for k = 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, ..., ; (A088359)

a(k)=2 for k = 1, 2, 7, 12, 14, 21, 24, 26, 29, 38, 42, 45, 47, 51, 53, ..., ; (1 followed by A266109)

a(k)=3 for k = 4, 15, 27, 30, 48, 54, 57, 61, 86, 96, 102, 105, 112, ..., ; (A267103)

a(k)=4 for k = 8, 31, 58, 62, 106, 116, 120, 125, 192, 212, 222, 226, ..., ;

a(k)=5 for k = 16, 63, 121, 126, 227, 242, 247, 253, 419, 454, 469, ..., ;

a(k)=6 for k = 32, 127, 248, 254, 475, 496, 502, 509, 894, 950, 971, ..., ;

a(k)=7 for k = 64, 255, 503, 510, 978, 1006, 1013, 1021, 1872, 1956, ..., ;

a(k)=8 for k = 128, 511, 1014, 1022, 1992, 2028, 2036, 2045, 3864, ..., ;

a(k)=9 for k = 256, 1023, 2037, 2046, 4029, 4074, 4083, 4093, 7893, ..., ;

a(k)=10 for k = 512, 2047, 4084, 4094, 8113, 8168, 8178, 8189, ..., . (End)

Compare above to array A265903. - Antti Karttunen, Jan 18 2016

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
• Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.
• Wikipedia, Hofstadter sequence
• Index entries for Hofstadter-type sequences

Crossrefs

Cf. A004001, A087686, A093879, A188163, A266109, A267103 and array A265903.

Cf. A088359 (positions of ones).

Cf. A265332 (essentially the same sequence, but with a(1) = 1 instead of 2).

Programs

Haskell

import Data.List (group)
a051135 n = a051135_list !! (n-1)
a051135_list = map length $ group a004001_list
-- Reinhard Zumkeller, Jun 03 2011

Magma

nmax:=200;
h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..5*nmax]]; // h = A004001
A188163:= function(n)
   for j in [1..3*nmax+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
A051135:= func< n | A188163(n+1) - A188163(n) >;
[A051135(n): n in [1..nmax]]; // G. C. Greubel, May 20 2024

Maple

a[1]:=1: a[2]:=1: for n from 3 to 300 do a[n]:=a[a[n-1]]+a[n-a[n-1]] od: A:=[seq(a[n],n=1..300)]:for j from 1 to A[nops(A)-1] do c[j]:=0: for n from 1 to 300 do if A[n]=j then c[j]:=c[j]+1 else fi od: od: seq(c[j],j=1..A[nops(A)-1]); # Emeric Deutsch, Jun 06 2006

Mathematica

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Array[a, 250]; Take[ Transpose[ Tally[t]][[2]], 105] (* Robert G. Wilson v, Jun 07 2011 *)

SageMath

@CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1)) # h=A004001
def A188163(n):
    for j in range(1,2*n+1):
        if h(j)==n: return j
def A051135(n): return A188163(n+1) - A188163(n)
[A051135(n) for n in range(1,201)] # G. C. Greubel, May 20 2024

Scheme

(define (A051135 n) (- (A188163 (+ 1 n)) (A188163 n))) ;; Antti Karttunen, Jan 18 2016

Formula

From Antti Karttunen, Jan 18 2016: (Start)

a(n) = A188163(n+1) - A188163(n). [after Reinhard Zumkeller's Jun 03 2011 comment above]

Other identities:

a(n) = 1 if and only if A093879(n-1) = 1. [See A188163 for a reason.]

(End)

Extensions

More terms from Jud McCranie

Added links (in parentheses) to recently submitted related sequences - Antti Karttunen, Jan 18 2016