A238477 a(n) = 32*n - 27 for n >= 1. Second column of triangle A238475.

5, 37, 69, 101, 133, 165, 197, 229, 261, 293, 325, 357, 389, 421, 453, 485, 517, 549, 581, 613, 645, 677, 709, 741, 773, 805, 837, 869, 901, 933, 965, 997, 1029, 1061, 1093, 1125, 1157, 1189, 1221, 1253, 1285, 1317, 1349, 1381, 1413, 1445, 1477, 1509, 1541, 1573

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 6 following the pattern udddd = 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 6*n - 5.

This appears in Example 2.1. for x = 4 in the M. Trümper paper given as a link below.

Examples

			a(1) = 5 because the Collatz sequence of length 6 is [5, 16, 8, 4, 2, 1], following the pattern udddd, ending in 1, and 5 is the smallest start number following this pattern ending in an odd number.
a(2) = 37 with the length 6 Collatz sequence [37, 112, 56, 28, 14, 7] ending in 12 - 5 = 7, and this is the second smallest start number with this sequence pattern 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. A017077 (first column), A238475, A239123 (third column).

Programs

Mathematica

CoefficientList[Series[(5 + 27 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

Formula

O.g.f.: x*(5+27*x)/(1-x)^2.

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

E.g.f.: 27 + exp(x)*(32*x - 27).

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