A233074 Numbers that are exactly midway between the nearest square and the nearest triangular number.

2, 5, 23, 32, 47, 52, 65, 86, 140, 161, 170, 193, 203, 228, 266, 312, 356, 389, 403, 438, 453, 490, 545, 610, 671, 716, 735, 782, 802, 851, 1007, 1085, 1142, 1166, 1250, 1311, 1503, 1598, 1667, 1696, 1767, 1870, 2098, 2177, 2210, 2291, 2325, 2408, 2528, 2792, 2883

Offset

1,1

Comments

Numbers k such that k = (s+t)/2, where s is the square nearest to k, t is the triangular number nearest to k, and s != t. If there are two nearest triangular numbers, either of them is acceptable. - Edited by Robert Israel, Oct 07 2019

The sequence of roots of nearest squares begins: 1, 2, 5, 6, 7, 7, 8, 9, 12, 13, 13, 14, 14, 15, 16, 18, 19, 20, 20, 21, 21, ...

The sequence of roots of nearest triangular numbers begins: 2, 3, 6, 7, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, ...

The sequence of k-t (equals s-k) begins: -1, -1, 2, 4, 2, -3, -1, -5, 4, 8, -1, 3, -7, -3, -10, 12, 5, 11, -3, 3, -12, -6, ...

Links

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

Example

5 is in the sequence because 6 and 4 are the triangular number and square nearest to 5, and 5 = (6+4)/2.
23 is in the sequence because 21 and 25 are the triangular number and square nearest to 23, and 23 = (21+25)/2.

Maple

f:= proc(y) local t, x, s, r, R;
   t:= y*(y+1)/2;
   R:= NULL;
   for x from ceil(sqrt(t))-1 to floor(sqrt(t))+1 do
     s:= x^2;
     if s = t then next
     elif s < t then if t-y > s then next fi
     else if t+y+1 < s then next fi
     fi;
     r:= (s+t)/2;
     if r::integer then R:= R, r fi
   od;
   R
end proc:
map(f, [$1..200]; # Robert Israel, Oct 06 2019

Mathematica

f[y_] := Module[{t, x, s, r, R = Nothing},
  t = y(y+1)/2;
  For[x = Ceiling[Sqrt[t]]-1, x <= Floor[Sqrt[t]]+1, x++,
    s = x^2;
    Which[s == t, Continue[], s < t,
    If[t - y > s, Continue[]], True,
    If[t + y + 1 < s, Continue[]]];
    r = (s + t)/2;
    If[IntegerQ[r], R = r]
  ];
R];
Map[f, Range[200]] (* Jean-François Alcover, Jul 30 2023, after Robert Israel *)

Prog

(Java)
import java.math.*;
public class A233074 {
  public static void main (String[] args) {
    for (long n = 1; ; n++) { // ok for small n
      long r2 = (long)Math.sqrt(n), b2 = r2*r2, a2 = (r2+1)*(r2+1);
      long t = (long)Math.sqrt(2*n), b3 = t*(t+1)/2, a3 = b3 + t + 1;
      if (b3 > n) {
        a3 = b3;
        b3 = t*(t-1)/2;
      }
      if ((b2+a3 == n*2 && n - b2 <= a2 - n && a3 - n <= n - b3) ||
          (b3+a2 == n*2 && n - b3 <= a3 - n && a2 - n <= n - b2))
            System.out.printf("%d, ", n);
    }
  }
}

Crossrefs

Cf. A000217, A000290, A233075.

Sequence in context: A038919 A141181 A191030 * A100031 A293213 A215278

Adjacent sequences: A233071 A233072 A233073 * A233075 A233076 A233077

Keyword

nonn,easy

Author

Alex Ratushnyak, Dec 03 2013

Extensions

Corrected by Alex Ratushnyak, Jun 08 2014

Status

approved