A175032 a(n) is the difference between the n-th triangular number and the next perfect square.

0, 0, 1, 3, 6, 1, 4, 8, 0, 4, 9, 15, 3, 9, 16, 1, 8, 16, 25, 6, 15, 25, 3, 13, 24, 36, 10, 22, 35, 6, 19, 33, 1, 15, 30, 46, 10, 26, 43, 4, 21, 39, 58, 15, 34, 54, 8, 28, 49, 0, 21, 43, 66, 13, 36, 60, 4, 28, 53, 79, 19, 45, 72, 9, 36, 64, 93, 26, 55, 85, 15, 45, 76, 3, 34, 66, 99, 22

Offset

0,4

Comments

All terms are from {0} U A175035. No terms are from A175034.

The sequence consists of ascending runs of length 3 or 4. The first run starts at n = 1 and thereafter the k-th run starts at n = A214858(k - 1). - John Tyler Rascoe, Nov 05 2022

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

a(n) = (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2. - Ctibor O. Zizka, Nov 09 2009

a(n) = A080819(n) - A000217(n). - R. J. Mathar, Aug 24 2010

Mathematica

Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[0, 80]] (* Harvey P. Dale, Aug 25 2013 *)

Prog

(PARI) a(n) = my(t=n*(n+1)/2); if (issquare(t), 0, (sqrtint(t)+1)^2 - t); \\ Michel Marcus, Nov 06 2022

Crossrefs

Cf. A001109, A214858.

Cf. A000217, A080819.

Cf. A175034, A175035.

Cf. sequences where a(m)=k: A001108 (0), A006451 (1), A154138 (3), A154139 (4), A154140 (6), A154141 (8), A154142 (9), A154143 (10), A154144 (13), A154145 (15), A154146 (16), A154147 (19), A154148 (21), A154149 (22), A154150(24), A154151 (25), A154151 (26), A154153(28), A154154 (30).

Sequence in context: A181099 A163327 A338460 * A078768 A307281 A089078

Adjacent sequences: A175029 A175030 A175031 * A175033 A175034 A175035

Keyword

easy,nonn,look

Author

Ctibor O. Zizka, Nov 09 2009

Extensions

Erroneous formula variant deleted and offset set to zero by R. J. Mathar, Aug 24 2010

Status

approved