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

1,2

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, ...

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A109152 A130469 A106381 * A174302 A108074 A189175

Adjacent sequences: A064781 A064782 A064783 * A064785 A064786 A064787

Keyword

nonn,look

Author

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

Extensions

Definition corrected by Harry J. Smith, Sep 25 2009

Terms corrected by Harry J. Smith, Sep 25 2009

Status

approved