OFFSET
0,3
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]
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)
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)
EXAMPLE
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
MATHEMATICA
Select[Range[121], Floor[Sqrt[#]]==Round[Sqrt[#]] &] (* Stefano Spezia, Oct 19 2024 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Floor van Lamoen, Jul 24 2001
STATUS
approved