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%I M3239 #43 Dec 26 2020 12:28:41
%S 1,4,5,6,9,12,15,16,17,20,21,22,25,26,27,30,33,36,37,38,41,44,47,48,
%T 49,52,55,58,59,60,63,64,65,68,69,70,73,76,79,80,81,84,85,86,89,90,91,
%U 94,97,100,101,102,105,106,107,110,111,112,115,118,121,122,123,126,129,132
%N A self-generating sequence (see Comments for definition).
%C From _N. J. A. Sloane_, Dec 26 2020: (Start)
%C The best definitions of the triple [this sequence, A003157, A003158] are as the rows a(n), b(n), c(n) of the table:
%C n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
%C a: 1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, ...
%C b: 3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, ...
%C c: 2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, ...
%C where a(1)=1, b(1)=3, c(1)=2, and thereafter
%C a(n) = mex{a(i), b(i), c(i), i<n},
%C b(n) = a(n) + 2*n,
%C c(n) = b(n) - 1.
%C Then a,b,c form a partition of the positive integers.
%C Note that there is another triple of sequences (A003144, A003145, A003146) also called a, b, c and also a partition of the positive integers, in a different paper by the same authors (Carlitz-Scovelle-Hoggatt) in the same volume of the same journal.
%C (End)
%C a(n) is the number of ones before the n-th zero in the Feigenbaum sequence A035263. - _Philippe Deléham_, Mar 27 2004
%C Number of odd numbers before the n-th even number in A007413, A007913, A001511, A029883, A033485, A035263, A036585, A065882, A065883, A088172, A092412. - _Philippe Deléham_, Apr 03 2004
%C Indices of a in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - _Philippe Deléham_, Apr 12 2004
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003156/b003156.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-1/carlitz3-a.pdf">Fibonacci representations of higher order</a>, Fib. Quart., 10 (1972), 43-69. [Here a, b, c are A003144, A003145, A003146.]
%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fibonacci Quarterly 10.5 (1972), 499-518, 550. [A003156, A003157, A003158 are on page 500.]
%F a(n) = A079523(n) - n + 1 = A003157(n) - 2n = A003158(n) - 2n + 1. - _Philippe Deléham_, Feb 28 2004
%F a(n) = A036554(n) - n = A072939(n) - n - 1 = 2*A003159(n) - n. - _Philippe Deléham_, Apr 10 2004
%F a(n) = Sum_{k = 1..n} A080426(k). - _Philippe Deléham_, Apr 16 2004
%p a:= proc(n) global l; while nops(l)<n do l:= map (d-> [1, 3$d, 1][], l) od; `if` (n=1, 1, a(n-1) +l[n]) end: l:= [1]: seq (a(n), n=1..80); # _Alois P. Heinz_, Oct 31 2009
%t Position[Nest[Flatten[# /. {0 -> {0, 2, 1}, 1 -> {0}, 2 -> {0}}]&, {0}, 7], 0] // Flatten (* _Jean-François Alcover_, Mar 14 2014 *)
%o (Haskell) following Deléham
%o a003156 n = a003156_list !! (n-1)
%o a003156_list = scanl1 (+) a080426_list
%o -- _Reinhard Zumkeller_, Oct 27 2014
%Y Cf. A001511, A003157, A003158, A003159, A007413, A007913, A029883, A033485, A035263, A036554, A036585, A056196, A065882, A065883, A072939, A079523, A072939, A080426, A088172, A092412, A092606.
%Y See also A003144, A003145, A003146.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Alois P. Heinz_, Oct 31 2009
%E Incorrect equation removed from formula by _Peter Munn_, Dec 11 2020
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