A386705 a(n) = sum of the 2^(n-1) even positive integers having bit length 2*n and in which, when written in binary, each run of 0's is of exactly the same length as the run of 1's immediately before it.

2, 22, 192, 1576, 12704, 101856, 815360, 6524032, 52194816, 417564160, 3340525568, 26724231168, 213793906688, 1710351376384, 13682811273216, 109462490742784, 875699927121920, 7005599419465728, 56044795360968704, 448358362898759680, 3586866903213146112, 28694935225753403392

Offset

1,1

Comments

Row sums of A166751, when viewed as an irregular triangle whose row terms have the same number of bits (see the Example section there).

Links

Table of n, a(n) for n=1..22.

Formula

Empirical: a(n) = 12*a(n-1) - 36*a(n-2) + 32*a(n-3), with a(1) = 2, a(2) = 22, a(3) = 192.

Example

For n = 3, the 2^(n-1) terms with bit length 2*n = 6 satisfying the criteria are (in binary): 101010, 101100, 110010 and 111000, corresponding (in decimal) to 42, 44, 50 and 56, giving a sum of 192.

Mathematica

A386705[n_] := With[{b = Array[IntegerDigits[4^# - 2^#, 2] &, n]}, Total[Map[FromDigits[Flatten[#], 2] &, Map[b[[#]] &, Map[Permutations, IntegerPartitions[n]], {2}], {2}], 2]];
Array[A386705, 20]

Crossrefs

Cf. A166751.

Sequence in context: A000184 A007613 A346796 * A279801 A043037 A058441

Adjacent sequences: A386702 A386703 A386704 * A386706 A386707 A386708

Keyword

nonn,base

Author

Paolo Xausa, Aug 28 2025

Status

approved