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%I #18 Aug 26 2015 16:58:36
%S 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,
%T 4,5,4,4,4,4,4,5,6,5,5,6,5,5,6,5,5,6,5,5,6,7,6,6,6,6,6,7,7,7,7,7,7,7,
%U 7,7,8,8,8,8,8,9,8,8,8,8,8,9,9,9,9,9,9,9,9,9,10,11,10,11,10,11,10,11,10,11,10,11,10,11,10,11,10,11,10,11,12,12,12,12,12,13,12,12,12,12
%N Parse the infinite string 0123456789012345678901234567890... into distinct phrases 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 01, 23, 45, 67, 89, 012, 34, 56, 78, 90, 12, 345, ...; a(n) = length of n-th phrase.
%C See A187180-A187187 for further details.
%C Answers a question raised by Sergio Verdu (personal communication, Mar 05 2011).
%H N. J. A. Sloane, <a href="/A187188/b187188.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_101">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
%F After the initial block of 10 1's, the sequence is quasi-periodic with period 100, increasing by 10 after each block. In more detail:
%F a(n) = 1 for 1 <= n <= 10.
%F For n >= 10, write n = 11 + 100i + j with i >= 0, 0 <= j <= 99.
%F Then for 0 <= j <= 79, a(n) = 10i + f(j),
%F where f(0) ... f(79) is the following 80-term sequence:
%F [2 2 2 2 2 3 2 2 2 2 2 3
%F 3 3 3 3 3 3 3 3
%F 4 4 4 4 4 5 4 4 4 4 4 5
%F 6 5 5 6 5 5 6 5 5 6 5 5
%F 6 7
%F 6 6 6 6 6
%F 7 7 7 7 7 7 7 7 7
%F 8 8 8 8 8 9 8 8 8 8 8 9
%F 9 9 9 9 9 9 9 9]
%F (this has been broken into blocks to make it easier to see),
%F and for 80 <= j <= 99, a(n) = 10i+10 if j is even, a(n) = 10i+11 if j is odd.
%F Examples:
%F n=120 = 11 + 100*1 + 9, i=1, j=9, a(120)=10+f(9) = 10+2 = 12
%F n=292 = 11 + 100*2 + 81, i=2, j=81. a(292)=20+11=31
%e The sequence begins
%e 1 1 1 1 1 1 1 1 1 1
%e 2 2 2 2 2 3 2 2 2 2 2 3
%e 3 3 3 3 3 3 3 3
%e 4 4 4 4 4 5 4 4 4 4 4 5
%e 6 5 5 6 5 5 6 5 5 6 5 5
%e 6 7
%e 6 6 6 6 6
%e 7 7 7 7 7 7 7 7 7
%e 8 8 8 8 8 9 8 8 8 8 8 9
%e 9 9 9 9 9 9 9 9
%e 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11
%e 12 12 12 12 12 13 12 12 12 12 12 13
%e ...
%Y See A187180-A187188 for alphabets of size 2 through 10.
%Y See also A109337, A187199, A187200.
%K nonn
%O 1,11
%A _N. J. A. Sloane_, Mar 06 2011
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