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

Author

Paolo Xausa, Aug 28 2025

Keywords

• nonn
• base
• new

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).

Examples

			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.
		

Crossrefs

Cf. A166751.

Programs

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]

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.