A309616 a(n) is the number of ways to represent 2*n in the decibinary system.

1, 2, 4, 6, 10, 13, 18, 22, 30, 36, 45, 52, 64, 72, 84, 93, 110, 122, 140, 154, 177, 192, 214, 230, 258, 277, 304, 324, 356, 376, 405, 426, 464, 490, 528, 557, 604, 634, 678, 710, 765, 802, 854, 892, 952, 989, 1042, 1080, 1146, 1190, 1253, 1300, 1374, 1420, 1486, 1533, 1612, 1664

Offset: 0

Author

Jonas Hollm, Aug 10 2019

Keywords

• nonn
• base

Comments

It appears that a(n) is the number of decibinary numbers that can be constructed to represent the decimal numbers 2n-2 and 2n-1. To make this more clear let's consider n = 5: a(5) = 10 means that there are 10 decibinary numbers that represent the decimal numbers 2*5 - 2 = 8 and 2*5 - 1 = 9.

Furthermore, a(n) is the number of k such that A028897(k)=2*n.

Examples

			a(1) = 1.
a(2) = a(2-1) + a(ceiling(2/2)) = a(1) + a(1) = 1 + 1 = 2.
a(3) = a(3-1) + a(ceiling(3/2)) = a(2) + a(2) = 2 + 2 = 4.
a(4) = a(4-1) + a(ceiling(4/2)) = a(3) + a(2) = 4 + 2 = 6.
a(5) = a(5-1) + a(ceiling(5/2)) = a(4) + a(3) = 6 + 4 = 10.
a(6) = a(6-1) + a(ceiling(6/2)) - a(ceiling((6-5)/2)) = a(5) + a(3) - a(1) = 10 + 4 - 1 = 13.
a(7) = a(7-1) + a(ceiling(7/2)) - a(ceiling((7-5)/2)) = a(6) + a(4) - a(1) = 13 + 6 - 1 = 18.
a(8) = a(8-1) + a(ceiling(8/2)) - a(ceiling((8-5)/2)) = a(7) + a(4) - a(2) = 18 + 6 - 2 = 22.
a(9) = a(9-1) + a(ceiling(9/2)) - a(ceiling((9-5)/2)) = a(8) + a(5) - a(2) = 22 + 10 - 2 = 30.
a(10) = a(10-1) + a(ceiling(10/2)) - a(ceiling((10-5)/2)) = a(9) + a(5) - a(3) = 30 + 10 - 4 = 36.
		

Links

• HackerRank, Decibinary Numbers.

Crossrefs

Cf. A007728: superseeker found that the deltas of the sequence a(n+1) - a(n) match transformations of the original query.

Cf. A028897.

Programs

Mathematica

Nest[Append[#1, #1[[-1]] + #1[[Ceiling[#2/2] ]] - If[#2 > 5, #1[[Ceiling[(#2 - 5)/2] ]], 0 ]] & @@ {#, Length@ # + 1} &, {1}, 57] (* Michael De Vlieger, Sep 29 2019 *)

Formula

a(1) = 1. a(n) = a(n-1) + a(ceiling(n/2)) if 1 < n <= 5.

Conjecture: a(n) = a(n-1) + a(ceiling(n/2)) - a(ceiling((n-5)/2)) if n > 5.

I think this sequence is closely related to the 10th binary partition function. The only difference is that every second number is omitted. At the moment, the 10th binary partition function is not in the OEIS. However, my experiments strongly suggest that the 10th binary partition function would indeed look like 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ...

Extensions

Name corrected by Rémy Sigrist, Oct 15 2019