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A239128 a(n) = 32*n - 1, n >= 1. Fourth column of triangle A239126, related to the Collatz problem. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.

Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

O.g.f.: x*(31+x)/(1-x)^2.

EXAMPLE

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.

MATHEMATICA

CoefficientList[Series[(31 + x)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)

32*Range[50]-1 (* Harvey P. Dale, Jan 25 2021 *)

CROSSREFS

Cf. A239126, A125169 (third column), A239127.

Sequence in context: A042904 A042902 A042900 * A042910 A042908 A042912

Adjacent sequences:  A239125 A239126 A239127 * A239129 A239130 A239131

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Mar 13 2014

STATUS

approved

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Last modified May 14 08:44 EDT 2021. Contains 343879 sequences. (Running on oeis4.)