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
a(n) = Sum_{k=0..n-1} 2^(k*(2*n-k-1)/2+n-1). - Chai Wah Wu, Jul 11 2025
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
(Python)
def A164894(n): return sum(1<<(k*((n<<1)-k-1)>>1)+n-1 for k in range(n)) # Chai Wah Wu, Jul 11 2025
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.
KEYWORD
base,easy,nonn
AUTHOR
Gil Broussard, Aug 29 2009
STATUS
approved
