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

Offset

1,1

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.

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.

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A238475, A238477 (second column).

Sequence in context: A259493 A253459 A041850 * A241697 A358994 A239569

Adjacent sequences: A239120 A239121 A239122 * A239124 A239125 A239126

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 10 2014

Status

approved