A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

1, 6, 52, 840, 26896, 1721376, 220336192, 56406065280, 28879905423616, 29573023153783296, 60565551418948191232, 248076498612011791288320, 2032242676629600594233921536, 33296264013899376135928570454016, 1091051979207454757222107396637212672

Offset

1,2

Comments

These numbers are half the sum of powers of 2 indexed by differences of a triangular number and each smaller triangular number (e.g., 21 - 15 = 6, 21 - 10 = 11, ..., 21 - 0 = 21).

This suggests another way to think about these numbers: consider the number triangle formed by the characteristic function of the triangular numbers (A010054), join together the first n rows (the very first row is row 0) as a single binary string and that gives the (n + 1)th term of this sequence. - Alonso del Arte, Nov 15 2013

Numbers k such that the k-th composition in standard order (row k of A066099) is an initial interval. - Gus Wiseman, Apr 02 2020

Links

Harvey P. Dale, Table of n, a(n) for n = 1..81

Formula

a(n) = Sum_{k=0..n-1} 2^((n^2 + n)/2 - (k^2 + k)/2 - 1). - Alonso del Arte, Nov 15 2013

Intersection of A333255 and A333217. - Gus Wiseman, Apr 02 2020

Example

a(1) = 1, also 1 in binary.
a(2) = 6, or 110 in binary.
a(3) = 52, or 110100 in binary.
a(4) = 840, or 1101001000 in binary.

Mathematica

Table[Sum[2^((n^2 + n)/2 - (k^2 + k)/2 - 1), {k, 0, n - 1}], {n, 25}] (* Alonso del Arte, Nov 14 2013 *)
Module[{nn=15, t}, t=Table[10^n, {n, 0, nn}]; Table[FromDigits[Flatten[IntegerDigits/@Take[t, k]], 2], {k, nn}]] (* Harvey P. Dale, Jan 16 2024 *)

Prog

(Python)
def a(n): return int("".join("1"+"0"*i for i in range(n)), 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jul 05 2021

Crossrefs

The version for prime (rather than binary) indices is A002110.

The non-strict generalization is A225620.

The reversed version is A246534.

Standard composition numbers of permutations are A333218.

Standard composition numbers of strict increasing compositions are A333255.

Cf. A000120, A029931, A048793, A066099, A070939, A124768, A233564, A272919, A333217, A333220, A333379.

Sequence in context: A271802 A097820 A166889 * A027835 A055973 A223345

Adjacent sequences: A164891 A164892 A164893 * A164895 A164896 A164897

Keyword

base,easy,nonn

Author

Gil Broussard, Aug 29 2009

Status

approved