A239128 a(n) = 32*n - 1, n >= 1. Fourth column of triangle A239126, related to the Collatz problem.

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, 1503, 1535, 1567, 1599

Offset: 1

Author

Wolfdieter Lang, Mar 13 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

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

Programs

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 *)

Formula

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

From Elmo R. Oliveira, Apr 04 2025: (Start)

E.g.f.: exp(x)*(32*x - 1) + 1.

a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)