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

%I A063656 #42 Oct 19 2024 22:07:15
%S A063656 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,
%T A063656 40,41,42,49,50,51,52,53,54,55,56,64,65,66,67,68,69,70,71,72,81,82,83,
%U A063656 84,85,86,87,88,89,90,100,101,102,103,104,105,106,107,108,109,110,121
%N A063656 Numbers k such that the truncated square root of k is equal to the rounded square root of k.
%C A063656 Also: take 1, skip 0, take 2, skip 1, take 3, skip 2, ...
%C A063656 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
%C A063656 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
%C A063656 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]
%H A063656 Seiichi Manyama, <a href="/A063656/b063656.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Harry J. Smith)
%F A063656 As a triangle from _Stefano Spezia_, Oct 19 2024: (Start)
%F A063656 T(n,k) = n^2 + k with 0 <= k <= n.
%F A063656 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)
%e A063656 The triangle begins as:
%e A063656    0;
%e A063656    1,  2;
%e A063656    4,  5,  6;
%e A063656    9, 10, 11, 12;
%e A063656   16, 17, 18, 19, 20;
%e A063656   25, 26, 27, 28, 29, 30;
%e A063656   36, 37, 38, 39, 40, 41, 42;
%e A063656   49, 50, 51, 52, 53, 54, 55, 56;
%e A063656   ... - _Stefano Spezia_, Oct 19 2024
%t A063656 Select[Range[121],Floor[Sqrt[#]]==Round[Sqrt[#]] &] (* _Stefano Spezia_, Oct 19 2024 *)
%o A063656 (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
%o A063656 (Haskell)
%o A063656 a063656 n = a063656_list !! n
%o A063656 a063656_list = f 1 [0..] where
%o A063656    f k xs = us ++ f (k + 1) (drop (k - 1) vs) where
%o A063656                     (us, vs) = splitAt k xs
%o A063656 -- _Reinhard Zumkeller_, Jun 20 2015
%Y A063656 Cf. A063657, A004201-A004202.
%Y A063656 Cf. A128217, A217575.
%Y A063656 Essentially partial sums of A051340.
%K A063656 nonn,tabl
%O A063656 0,3
%A A063656 _Floor van Lamoen_, Jul 24 2001