A128549 -id:A128549 - cp's OEIS frontend

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

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

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, ...

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

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

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

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

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 *)

{ 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

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

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

Definition corrected by Harry J. Smith, Sep 25 2009

Terms corrected by Harry J. Smith, Sep 25 2009

A373327 Records of A064784.

Original entry on oeis.org

0, 2, 6, 9, 14, 20, 21, 28, 35, 44, 51, 57, 62, 75, 80, 84, 100, 104, 107, 126, 129, 131, 132, 155, 156, 182, 209, 237, 266, 291, 321, 344, 375, 396, 428, 461, 480, 514, 531, 566, 602, 617, 654, 667, 705, 744, 755, 795, 804, 845, 852, 894, 937, 942, 986, 989, 1034

Offset: 1

Hugo Pfoertner, Jun 01 2024

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

A373328 gives the corresponding positions.

Cf. A000217, A000290, A128549.

R:= 0: m:= 0: count:= 1:
for i from 1 while count < 100 do
  t:= i*(i+1)/2;
  v:= t - floor(sqrt(t))^2;
  if v > m then
    R:= R,v; m:= v; count:= count+1;
  fi
od:
R; # Robert Israel, Dec 19 2024

a373327(nmax) = {my(m=-oo); for(n=0, nmax, my(T=n*(n+1)/2, d=T-sqrtint(T)^2); if(d>m, print1(d, ", "); m=d))};
a373327(750)

A373328 Numbers k such that the difference between T = k*(k+1)/2 = A000217(k) and the largest square <= T reaches a new record.

Original entry on oeis.org

0, 2, 5, 9, 12, 15, 19, 22, 29, 32, 39, 46, 53, 56, 63, 70, 73, 80, 87, 90, 97, 104, 111, 114, 121, 131, 155, 172, 189, 213, 230, 254, 271, 295, 312, 329, 353, 370, 394, 411, 428, 452, 469, 493, 510, 527, 551, 568, 592, 609, 633, 650, 667, 691, 708, 732, 749, 766

Offset: 1

Hugo Pfoertner, Jun 01 2024

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

A373327 gives the corresponding differences.

Cf. A000217, A000290, A064784, A128549.

J:= 0: m:= 0: count:= 1:
for i from 1 while count < 100 do
  t:= i*(i+1)/2;
  v:= t - floor(sqrt(t))^2;
  if v > m then
    J:= J,i; m:= v; count:= count+1;
   fi
od:
J; # Robert Israel, Dec 19 2024

a373328(nmax) = {my(m=-oo); for(n=0, nmax, my(T=n*(n+1)/2, d=T-sqrtint(T)^2); if(d>m, print1(n, ", "); m=d))};
a373328(770)

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

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A373327 Records of A064784.

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A373328 Numbers k such that the difference between T = k*(k+1)/2 = A000217(k) and the largest square <= T reaches a new record.

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Maple

PARI