A338546 For n > 0, a(n) is the number of 1's among the first T(n) terms of the sequence 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, ..., k 1's, k 0's, where T(n) is the n-th triangular number.

1, 2, 3, 6, 9, 11, 15, 21, 24, 28, 36, 42, 46, 55, 65, 70, 78, 91, 99, 105, 120, 133, 140, 153, 171, 180, 190, 210, 225, 234, 253, 275, 285, 300, 325, 341, 352, 378, 402, 414, 435, 465, 481, 496, 528, 553, 567, 595, 630, 645, 666, 703, 728, 744, 780, 816

Offset: 1

Author

Arnauld Chevallier, Nov 02 2020

Keywords

• nonn

Comments

The sequence of 1's and 0's is more formally defined as A118175.

Examples

			Equivalently, this is the number of 1's up to the n-th row of a triangle filled with A118175:
   1 (1)
   01 (2)
   100 (3)
   1110 (6)
   00111 (9)
   100001 (11)
   1111000 (15)
   00111111 (21)
   000000111 (24)
   1111000000 (28)
   01111111100 (36)
   000000111111 (42)
   ...
		

Crossrefs

Based on A118175 and A000217.

Programs

Mathematica

With[{nn = 56}, Accumulate@ Map[Total, TakeList[Array[1 - Ceiling[#] + Round[#] &@ Sqrt[#] &, PolygonalNumber[nn]], Range[nn]]]] (* Michael De Vlieger, Nov 02 2020, after Branko Curgus at A118175 *)

PARI

f(n) = my(r=sqrt(n)); 1 - ceil(r) + round(r);
a(n) = sum(k=1, n*(n+1)/2, f(k)==1); \\ Michel Marcus, Nov 12 2020