OFFSET
1,1
COMMENTS
Here xy^k means the concatenation of the words x and k copies of y.
The first '5' appears at step 343. At which step does the first '6' appear? - Sergio Pimentel, Jul 13 2015
The first '6' appears at about 4.33*10^616, see FORMULA for exact value. The positions where record lengths of strings 5 and 6 occur can be computed via sequence A091844, see PARI code there. - M. F. Hasler, Sep 29 2018
This sequence is the concatenation of the "glue strings" of sequence A091787, just like sequence A091844 is the concatenation of the glue strings of this sequence. - M. F. Hasler, Oct 04 2018
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
Simon Plouffe, On the values of the functions zeta and gamma, arXiv preprint arXiv:1310.7195 [math.NT], 2013.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
FORMULA
From M. F. Hasler, Sep 29 2018: (Start)
The first '5' appears at position 343 which in hexadecimal is 1x, with x = 57 (in base 16).
The first two consecutive '5's appear at positions 5760309085..5760309086, which is {-1..0} + 1y in hexadecimal, where y = xxxx + 7.
Then '55' occurs again at positions {-1..0} + 1yy in hexadecimal, etc.
The first three consecutive '5's appear at positions 1z + {-2..0} in hexadecimal, where z = yyyy + 20 (in base 16).
Then '555' occurs again at positions {-2..0} + 1zz in hexadecimal, etc.
The first occurrence of '5555' is at positions {-4..-1} + 1w in hexadecimal, where w = zzzz + 98 (in base 16).
'555' also occurs again at positions {-2..0} + 1w'z in hexadecimal, where 'z = 'yyyy + 20, 'y = 'xxxx + 7 and 'x = x - 1 = 56 (in base 16), and also at positions {-2..0} + 1w'zz in hexadecimal, and at positions {-2..0} + 1w'zzz in hexadecimal.
Then '5555' occurs again at positions {-4..-1} + 1w'w in hexadecimal, where 'w = 'zzzz + 98 (in base 16).
'555' also occurs again at positions {-2..0} + 1w'w'z in hexadecimal, etc.
'5555' also occurs again at positions {-3..0} + 1w'w'w in hexadecimal, and at positions {-4..-1} + 1w'w'w'w in hexadecimal.
The first occurrence of a '6' (immediately after the first '55555', followed by a sixth '5') is at position 1w'w'w'w + 30E in hexadecimal, which equals 782 + 16^510*257 + (16^512-1)/(16^128-1)*(158 - 16^126 + (16^128-1)/(16^32-1)*(32 + (16^32-1)/(16^8-1)*(7 + (16^8-1)/255*87))). (End)
MATHEMATICA
maxBlockLength = 21; a[1] = 3; a[n_] := a[n] = Module[{rev = Reverse[Array[a, n-1]]}, blockCount[blockLength_] := Module[{par, p1, k}, par = Partition[rev, blockLength]; If[par == {}, Return[1]]; p1 = First[par]; k = 1; While[k <= Length[par], If[par[[k]] != p1, Break[], k++]]; k - 1]; Max[Max[Array[blockCount, maxBlockLength]], 3]]; Array[a, 99] (* Jean-François Alcover, Nov 07 2013 *)
PROG
(PARI) A091799(n, A=[])={while(#A<n, my(k=3, L=0, m=k); while((k+1)*(L+1)<=#A, for(N=L+1, #A/(m+1), A[-m*N..-1]==A[-(m+1)*N..-N-1]&& m++&& break); m>k||break; k=m); A=concat(A, k)); A} \\ M. F. Hasler, Aug 07 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 08 2004
STATUS
approved