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 A000194 n appears 2n times; also nearest integer to square root of n. 36

%I

%S 0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,

%T 6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,

%U 8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10

%N n appears 2n times; also nearest integer to square root of n.

%C a(n) = inverse (frequency distribution) sequence of A002378(n-1). - _Jaroslav Krizek_, Jun 14 2009

%C Define the oblong root obrt(x) to be the (larger) solution of y * (y+1) = x; i.e., obrt(x) = sqrt(x+1/4) - 1/2. So obrt(x) is an integer iff x is an oblong number (A002378). Then a(n) = ceiling(obrt(n)). - _Franklin T. Adams-Watters_, Jun 24 2015

%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.

%H T. D. Noe, <a href="/A000194/b000194.txt">Table of n, a(n) for n = 0..1000</a>

%H G. Gutin, <a href="http://dx.doi.org/10.1016/j.disc.2007.07.021">Problem 913 (BCC20.5)</a>, Mediated digraphs, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.

%H M. A. Nyblom, <a href="http://www.jstor.org/stable/2695446">Some curious sequences involving floor and ceiling functions</a>, Am. Math. Monthly 109 (#6, 2002), 559-564.

%H M. Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: f(x^2, x^6) * x / (1 - x) where f(,) is Ramanujan's two-variable theta function. - _Michael Somos_, May 31 2000

%F a(n) = a(n-2*a(n-a(n-1)))+1. - _Benoit Cloitre_, Oct 27 2002

%F a(n+1) = a(n) + A005369(n).

%F a(n) = floor((1/2)*(1 + sqrt(4*n - 3))). - _Zak Seidov_, Jan 18 2006

%F a(n) = A000037(n) - n. - _Jaroslav Krizek_, Jun 14 2009

%F a(n) = floor(A027434(n)/2). - _Gregory R. Bryant_, Apr 17 2013

%F From _Mikael Aaltonen_, Jan 17 2015: (Start)

%F a(n) = floor(sqrt(n)+1/2).

%F a(n) = sqrt(A053187(n)). (End)

%e G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 3*x^10 + ...

%p Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=1..100) ];

%t A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* _Enrique PĂ©rez Herrero_, Apr 14 2010 *)

%t Flatten[Table[PadRight[{},2n,n],{n,10}]] (* _Harvey P. Dale_, Nov 16 2011 *)

%o (PARI) {a(n) = ceil( sqrtint(4*n) / 2)}; /* _Michael Somos_, Feb 11 2004 */

%o a000194 n = a000194_list !! (n-1)

%o a000194_list = concat \$ zipWith (\$) (map replicate [2,4..]) [1..]

%o -- _Reinhard Zumkeller_, Mar 18 2011

%Y Partial sums of A005369.

%Y A000037(n) - n.

%Y Cf. A002024, A259351, A002378.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_