A348624 a(n) = sum of row n of A348433 expressed as an irregular triangle.

31, 21, 2555, 2805, 3315, 17391, 38893, 104857575, 59363, 2097120, 31713, 376809, 117440484, 18790481885, 197132241, 2885681109, 42991575, 4966055899, 13153337295, 3959422917, 120946279055305, 4191888080835, 3729543441416139, 321057395310519, 84662395338675, 294669116243901

Offset: 1

Author

Michael De Vlieger, Oct 25 2021

Keywords

• nonn
• base

Comments

The binary expansion w of a(n) has an interesting appearance shown by the bitmap in links. We may divide w with length m into 3 parts: the most significant part includes all bits including the last 0 before the middle of the word, m/2, a central run of k 1's that includes all but the last 1 before a 0, and a least significant part that includes the last 1 in the central run of 1s and an assortment of 0's. For example, a(3) = 2555 -> 100.11111.1011, which we may partition as shown by "." so as to preserve the otherwise-leading 0 in the last part. The central run of 1s generally increases in length as n increases.

Examples

			Table showing the first 5 rows of A348433 each having A348408(n) terms, and their sum a(n):
n\k 1  2  3   4   5   6   7    8    9     a(n)   binary(a(n))
--------------------------------------------------------------
1:  1  2  4   8  16                   ->   31 ->        11111
2:  7 14                              ->   21 ->        10101
3:  5 10 20  40  80 160 320  640 1280 -> 2555 -> 100111111011
4: 11 22 44  88 176 352 704 1408      -> 2805 -> 101011110101
5: 13 26 52 104 208 416 832 1664      -> 3315 -> 110011110011
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1540
• Michael De Vlieger, Log log scatterplot of a(n) for n=1..120.
• Michael De Vlieger, Log log scatterplot of a(n) for n=1..3961.
• Michael De Vlieger, Big-endian bitmap of the binary expansion of a(n) for n=1..3961 showing 1s in black and 0s in white.

Crossrefs

Cf. A348408, A348433.

Programs

Mathematica

c[1] = m = q = 1; Most@ Reap[Do[If[IntegerQ[c[#]], Set[n, 2 m], Set[n, #]] &@ Total@ IntegerDigits[m]; If[m > n, Sow[q]; Set[q, n], q += n]; Set[c[n], 1]; m = n, 650]][[-1, -1]]
(* Extract up to 3961 terms from bitmap: *)
Block[{s = ImageData[ColorNegate@ Import["https://oeis.org/A348624/a348624_2.png"], "Bit"]}, Array[FromDigits[s[[#]], 2] &, 26]] (* Michael De Vlieger, Oct 26 2021 *)