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A051135 a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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, ..., ; etc. - Robert G. Wilson v, Jun 07 2011
Compare above to array A265903. - Antti Karttunen, Jan 18 2016
LINKS
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.
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)
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 *)
PROG
(Haskell)
import Data.List (group)
a051135 n = a051135_list !! (n-1)
a051135_list = map length $ group a004001_list
-- Reinhard Zumkeller, Jun 03 2011
(Scheme) (define (A051135 n) (- (A188163 (+ 1 n)) (A188163 n))) ;; Antti Karttunen, Jan 18 2016
CROSSREFS
Cf. A088359 (positions of ones).
Cf. A265332 (essentially the same sequence, but with a(1) = 1 instead of 2).
Sequence in context: A355522 A268190 A241150 * A325541 A260258 A283196
KEYWORD
easy,nonn,nice
AUTHOR
Robert Lozyniak (11(AT)onna.com)
EXTENSIONS
More terms from Jud McCranie
Added links (in parentheses) to recently submitted related sequences - Antti Karttunen, Jan 18 2016
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)