A239123 a(n) = 128*n - 107 for n >= 1. Third column of triangle A238475.

21, 149, 277, 405, 533, 661, 789, 917, 1045, 1173, 1301, 1429, 1557, 1685, 1813, 1941, 2069, 2197, 2325, 2453, 2581, 2709, 2837, 2965, 3093, 3221, 3349, 3477, 3605, 3733, 3861, 3989, 4117, 4245, 4373, 4501, 4629, 4757, 4885, 5013, 5141, 5269, 5397, 5525, 5653

Offset: 1

Author

Wolfdieter Lang, Mar 10 2014

Keywords

• nonn
• easy

Comments

This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 8 following the pattern ud^6 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 Collatz sequence is required to be odd, and it is given by 6*n - 5.

This appears in Example 2.1. for x = 6 in the M. Trümper paper given as a link below.

Examples

			a(1) = 21 because the Collatz sequence of length 8 is [21, 64, 32, 16, 8, 4, 2, 1] ending in 6*1-5 = 1, and 21 is the smallest positive number following this pattern udddddd ending in an odd number.
a(2) = 149 with the length 8 Collatz sequence [149, 448, 224, 112, 56, 28, 14, 7] ending in 6*2 - 5 = 7, and 149 is the second smallest start number following this pattern ud^6, ending in an odd 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. A238475, A238477 (second column).

Programs

Mathematica

CoefficientList[Series[(21 + 107 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

Formula

O.g.f.: x*(21+107*x)/(1-x)^2.

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

E.g.f.: 107 + exp(x)*(128*x - 107).

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