

A091799


a(1) = 3. To get a(n+1), write the string a(1)a(2)...a(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far. Then a(n+1) = max(k,3).


9



3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3
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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



FORMULA

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 + (16^5121)/(16^1281)*(158  16^126 + (16^1281)/(16^321)*(32 + (16^321)/(16^81)*(7 + (16^81)/255*87))). (End)


MATHEMATICA

maxBlockLength = 21; a[1] = 3; a[n_] := a[n] = Module[{rev = Reverse[Array[a, n1]]}, 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] (* JeanFrançois 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..N1]&& m++&& break); m>kbreak; k=m); A=concat(A, k)); A} \\ M. F. Hasler, Aug 07 2018


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



