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A233074 Numbers that are exactly midway between the nearest square and the nearest triangular number. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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,changed

AUTHOR

Alex Ratushnyak, Dec 03 2013

EXTENSIONS

Corrected by Alex Ratushnyak, Jun 08 2014

STATUS

approved

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Last modified October 18 07:41 EDT 2019. Contains 328146 sequences. (Running on oeis4.)