A064784 Difference between n-th triangular number t(n) and the largest square <= t(n).

0, 2, 2, 1, 6, 5, 3, 0, 9, 6, 2, 14, 10, 5, 20, 15, 9, 2, 21, 14, 6, 28, 20, 11, 1, 27, 17, 6, 35, 24, 12, 44, 32, 19, 5, 41, 27, 12, 51, 36, 20, 3, 46, 29, 11, 57, 39, 20, 0, 50, 30, 9, 62, 41, 19, 75, 53, 30, 6, 66, 42, 17, 80, 55, 29, 2, 69, 42, 14, 84, 56, 27, 100, 71, 41, 10, 87, 56

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 20 2001

Keywords

• nonn
• look

Comments

The second differences of a(n) - (a(n)-a(n-1))-(a(n-1)-a(n-2)) - give 2, -2, -1, 6, -6, -1, -1, 12, -12, -1, 16, -16, -1 ... 82k+2, 82k-2, -1, 82k+6, 82k-6, -1, -1, 82k+12, 82k-12, -1, 82k+16, -82k-16, -1, 82k+20, -82k-20, -1, -1, 82k+26, -82k-26, -1, 82k+30, -82k-30, -1, -1, 82k+36, -82k-36, -1, 82k+40, -82k-40, -1, 82k+44, -82k-44, -1, -1, 82k+50, -82k-50, -1, 82k+54, -82k-54, -1, -1, 82k+60, -82k-60, -1, 82k+64, -82k-64, -1, -1, 82k+70, -82k-70, -1, 82k+74, -82k-74, -1, 82k+78, -82k-78, -1, -1, ...

Examples

			n = 5: A000217(5) = 28, largest square below that is 25, so a(5) = 28 - 25 = 3.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
• J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
• Index entries for sequences related to EKG sequence

Crossrefs

Cf. A001108, A076816, A128549, A230038. Unique values are in A230044.

Programs

Maple

seq(n*(n+1)/2-floor(sqrt(n*(n+1)/2))^2,n=0..100);

Mathematica

f[n_]:=n*(n+1)/2-Floor[Sqrt[n*(n+1)/2]]^2; lst={}; Do[AppendTo[lst,f[n]],{n,0,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 17 2010 *)
#-Floor[Sqrt[#]]^2&/@Accumulate[Range[100]] (* Harvey P. Dale, Oct 15 2014 *)

PARI

{ default(realprecision, 100); for (n=1, 1000, t=n*(n + 1)/2; a=t - floor(sqrt(t))^2; write("b064784.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 25 2009

Python

from math import isqrt
def A064784(n): return (m:=n*(n+1)>>1)-isqrt(m)**2 # Chai Wah Wu, Jun 01 2024

Formula

a(n) = n*(n+1)/2 - floor(sqrt(n*(n+1)/2))^2.

a(n) = A053186(A000217(n)). - R. J. Mathar, Sep 10 2016

a(A001108(n)) = 0. - Hugo Pfoertner, Jun 01 2024

Extensions

Definition corrected by Harry J. Smith, Sep 25 2009

Terms corrected by Harry J. Smith, Sep 25 2009