A175032 - OEIS

%I #39 Nov 12 2022 08:29:35

%S 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,

%T 35,6,19,33,1,15,30,46,10,26,43,4,21,39,58,15,34,54,8,28,49,0,21,43,

%U 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

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

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

%C 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

%H Seiichi Manyama, <a href="/A175032/b175032.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2. - _Ctibor O. Zizka_, Nov 09 2009

%F a(n) = A080819(n) - A000217(n). - _R. J. Mathar_, Aug 24 2010

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

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

%Y Cf. A001109, A214858.

%Y Cf. A000217, A080819.

%Y Cf. A175034, A175035.

%Y 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).

%K easy,nonn,look

%O 0,4

%A _Ctibor O. Zizka_, Nov 09 2009

%E Erroneous formula variant deleted and offset set to zero by _R. J. Mathar_, Aug 24 2010