A239124 a(n) = 64*n - 11 for n >= 1. Third column of triangle A238476.

53, 117, 181, 245, 309, 373, 437, 501, 565, 629, 693, 757, 821, 885, 949, 1013, 1077, 1141, 1205, 1269, 1333, 1397, 1461, 1525, 1589, 1653, 1717, 1781, 1845, 1909, 1973, 2037, 2101, 2165, 2229, 2293, 2357, 2421, 2485, 2549, 2613, 2677, 2741, 2805, 2869, 2933, 2997

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 7 following the pattern ud^5 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, requiring that the sequence ends in an odd number. The last entry of this Collatz sequence is 6*n - 1.

This appears in Example 2.1. for x = 5 in the M. Trümper paper given as a link below.

Examples

			a(1) = 53 because the Collatz sequence of length 7 following the pattern uddddd, ending in an odd number is [53, 160, 80, 40, 20, 10, 5]. The end number is 6*1 - 1 = 5.
		

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. A004767 (first column), A082285 (second column), A238476.

Programs

Magma

[64*n-11: n in [1..50]]; // Vincenzo Librandi, Mar 13 2014

Mathematica

CoefficientList[Series[(53 + 11 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)
64 Range[40]-11 (* Harvey P. Dale, Nov 21 2018 *)

Formula

O.g.f.: x*(53+11*x)/(1-x)^2.

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

E.g.f.: 11 + exp(x)*(64*x - 11).

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