A239129 a(n) = 18*n - 1, n >= 1, the second column of triangle A239127 related to the Collatz problem.

17, 35, 53, 71, 89, 107, 125, 143, 161, 179, 197, 215, 233, 251, 269, 287, 305, 323, 341, 359, 377, 395, 413, 431, 449, 467, 485, 503, 521, 539, 557, 575, 593, 611, 629, 647, 665, 683, 701, 719, 737, 755, 773, 791, 809, 827, 845, 863, 881, 899, 917, 935, 953, 971

Offset: 1

Author

Wolfdieter Lang, Mar 13 2014

Keywords

• nonn
• easy

Comments

This sequence gives all ending values a(n) (in increasing order) of Collatz sequences of length 5 following the pattern (ud)^2, 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. The first entry is also odd and is given by M(2,n) = 8*n-1 from the array A239126.

This appears as N in Example 2.2. for x=y = 2 in the M. Trümper paper on p. 7, given as a link below.

Examples

			a(1) = 17 because the Collatz sequence for M(2,1) = 8*1 - 1 = 7 from A239126 is [7, 22, 11, 34, 17] ending in the odd number 17.
a(4) = 71 with the Collatz sequence of length 5 starting with M(2,4) = 31 given by [31, 94, 47, 142, 71], ending in a(4).
		

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. A016969 (first column), A239126, A239127.

Programs

Mathematica

CoefficientList[Series[(x + 17)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)

Formula

a(n) = 18*n - 1 for n >= 1.

O.g.f.: x*(x+17)/(1-x)^2.

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

E.g.f.: exp(x)*(18*x - 1) + 1.

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