

A164894


Base10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.


16



1, 6, 52, 840, 26896, 1721376, 220336192, 56406065280, 28879905423616, 29573023153783296, 60565551418948191232, 248076498612011791288320, 2032242676629600594233921536, 33296264013899376135928570454016, 1091051979207454757222107396637212672
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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 kth composition in standard order (row k of A066099) is an initial interval.  Gus Wiseman, Apr 02 2020


LINKS

Table of n, a(n) for n=1..15.


FORMULA

a(n) = Sum_{k=0..n1} 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 *)


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 nonstrict 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



