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A003156
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A self-generating sequence (see Comments for definition).
(Formerly M3239)
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10
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1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 33, 36, 37, 38, 41, 44, 47, 48, 49, 52, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 73, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 97, 100, 101, 102, 105, 106, 107, 110, 111, 112, 115, 118, 121, 122, 123, 126, 129, 132
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OFFSET
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1,2
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COMMENTS
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The best definitions of the triple [this sequence, A003157, A003158] are as the rows a(n), b(n), c(n) of the table:
n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
a: 1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, ...
b: 3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, ...
c: 2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, ...
where a(1)=1, b(1)=3, c(1)=2, and thereafter
a(n) = mex{a(i), b(i), c(i), i<n},
b(n) = a(n) + 2*n,
c(n) = b(n) - 1.
Then a,b,c form a partition of the positive integers.
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.
(End)
a(n) is the number of ones before the n-th zero in the Feigenbaum sequence A035263. - Philippe Deléham, Mar 27 2004
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
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. [Here a, b, c are A003144, A003145, A003146.]
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FORMULA
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MAPLE
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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
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MATHEMATICA
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Position[Nest[Flatten[# /. {0 -> {0, 2, 1}, 1 -> {0}, 2 -> {0}}]&, {0}, 7], 0] // Flatten (* Jean-François Alcover, Mar 14 2014 *)
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PROG
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(Haskell) following Deléham
a003156 n = a003156_list !! (n-1)
a003156_list = scanl1 (+) a080426_list
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CROSSREFS
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Cf. A001511, A003157, A003158, A003159, A007413, A007913, A029883, A033485, A035263, A036554, A036585, A056196, A065882, A065883, A072939, A079523, A072939, A080426, A088172, A092412, A092606.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Incorrect equation removed from formula by Peter Munn, Dec 11 2020
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STATUS
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approved
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