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

1,2

Comments

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

Links

Table of n, a(n) for n=1..56.

Example

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)
   ...

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 *)

Prog

(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

Crossrefs

Based on A118175 and A000217.

Sequence in context: A097384 A067886 A227000 * A358130 A176189 A057127

Adjacent sequences: A338543 A338544 A338545 * A338547 A338548 A338549

Keyword

nonn

Author

Arnauld Chevallier, Nov 02 2020

Status

approved