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A239128
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a(n) = 32*n - 1, n >= 1. Fourth column of triangle A239126, related to the Collatz problem.
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2
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31, 63, 95, 127, 159, 191, 223, 255, 287, 319, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, 831, 863, 895, 927, 959, 991, 1023, 1055, 1087, 1119, 1151, 1183, 1215, 1247, 1279, 1311, 1343, 1375, 1407, 1439, 1471
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OFFSET
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1,1
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COMMENTS
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This sequence gives all starting values a(n) (in increasing order) of Collatz sequences of length 9 following the pattern (ud)^4, with u (for `up'), mapping an odd number m to 3*m+1, and d (for `down'), mapping an even number m to m/2. The last entry of this sequence is required to be odd and it is given by 162*n-1.
This appears in Example 2.2. for x=y = 4 in the M. Trümper paper on p. 7, given as a link below.
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LINKS
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FORMULA
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O.g.f.: x*(31+x)/(1-x)^2.
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EXAMPLE
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a(1) = 31 because the Collatz sequence following the pattern udududud is [31, 94, 47, 142, 71, 214, 107, 322, 161], with length 9, ending in the odd number N(4,1) = 161 = 162*1 - 1 from the array A239127, and 31 is the smallest positive number whose Collatz sequence follows this pattern and ends in an odd number.
a(4) = 127 with the Collatz sequence [127, 382, 191, 574, 287, 862, 431, 1294, 647] ending in N(4,4) = 647 = 32*4 - 1. 127 is the fourth smallest positive number following this pattern with odd end number.
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MATHEMATICA
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CoefficientList[Series[(31 + x)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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