A063656 Numbers k such that the truncated square root of k is equal to the rounded square root of k.

0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 16, 17, 18, 19, 20, 25, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 41, 42, 49, 50, 51, 52, 53, 54, 55, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121

Offset: 0

Author

Floor van Lamoen, Jul 24 2001

Keywords

• nonn
• tabl

Comments

Also: take 1, skip 0, take 2, skip 1, take 3, skip 2, ...

The union of sets of numbers in closed intervals [k^2,k^2+k], k >= 0, intervals 0 to 1, 1 to 2, 4 to 6, 9 to 12 etc. - J. M. Bergot, Jun 27 2013

Conjecture: the following definition produces a(n) for n >= 1: a(1) = 1; for n > 1, smallest number > a(n-1) satisfying the condition that a(n) is a square if and only if n is a triangular number. - J. Lowell, May 13 2014

Thus a(2) = 2, because 2 is not a triangular number and not a square; a(3) != 3, because 3 is not a square but is a triangular number; a(3) = 4 is OK because 4 is a square and 3 is a triangular number; etc. [Examples supplied by N. J. A. Sloane, May 13 2014]

Examples

			The triangle begins as:
   0;
   1,  2;
   4,  5,  6;
   9, 10, 11, 12;
  16, 17, 18, 19, 20;
  25, 26, 27, 28, 29, 30;
  36, 37, 38, 39, 40, 41, 42;
  49, 50, 51, 52, 53, 54, 55, 56;
  ... - _Stefano Spezia_, Oct 19 2024
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Crossrefs

Cf. A063657, A004201-A004202.

Cf. A128217, A217575.

Essentially partial sums of A051340.

Programs

Haskell

a063656 n = a063656_list !! n
a063656_list = f 1 [0..] where
   f k xs = us ++ f (k + 1) (drop (k - 1) vs) where
                    (us, vs) = splitAt k xs
-- Reinhard Zumkeller, Jun 20 2015

Mathematica

Select[Range[121],Floor[Sqrt[#]]==Round[Sqrt[#]] &] (* Stefano Spezia, Oct 19 2024 *)

PARI

{ n=-1; for (m=0, 10^9, if (sqrt(m)%1 < .5, write("b063656.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 27 2009

Formula

As a triangle from Stefano Spezia, Oct 19 2024: (Start)

T(n,k) = n^2 + k with 0 <= k <= n.

G.f.: x*(1 + x + 2*y - 4*x*y + 3*x^3*y^2 - x^2*y*(2 + y))/((1 - x)^3*(1 - x*y)^3). (End)