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%I #127 Jan 10 2024 23:59:18
%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, for n >= 1; also nearest integer to square root of n.
%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
%C From _Wolfdieter Lang_, Mar 12 2019: (Start)
%C The general Pell equation is related to the non-reduced form F(n) = Xvec^T A(n) Xvec = x^2 - D(n)*y^2 with D(n) = A000037(n) (D not a square), Xvec = (x,y)^T (T for transposed) and A(n) = matrix[[1,0], [0,-D(n)]]. The discriminant of F(n) = [1, 0, -D(n)] is 4*D(n).
%C The first reduced form appears after two applications of an equivalence transformation A' = R^T A R obtained with R = R(t) = matrix([0, -1], [1, t]), namely first with t = 0, leading to the still not reduced form [-D, 0, 1], and then with t = ceiling(f(4*D(n))/2 - 1), where f(4*D(n)) = ceiling(2*sqrt(D(n))). This can be shown to be a(n), which is also D(n) - n, for n >= 1 (see a formula below).
%C This leads to the reduced form FR(n) = [1, 2*a(n), -(D(n) - a(n)^2)] = [1, 2*a(n), -(n - a(n)*(a(n) - 1))]. Example: n = 5, a(5) = 2: D(5) = 7 and FR(5) = [1, 4, -3]. (End)
%D Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 59-60.
%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 Jonathan M. Borwein and others, <a href="https://www.jstor.org/stable/2975334">Nearest Integer Zeta Functions, solution to Problem 10212</a>, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.
%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 Michael 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 a(n) = A000037(n) - n.
%F G.f.: x * f(x^2, x^6)/(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)
%F a(0) = 0, and a(n) = k for k from the closed interval [k^2 - k + 1, k*(k+1)] = [A002061(k), A002378(k)], for k >= 1. See A053187. - _Wolfdieter Lang_, Mar 12 2019
%F a(n) = floor(2*sqrt(n)) - floor(sqrt(n)). - _Ridouane Oudra_, Jun 08 2020
%F Sum_{n>=1} 1/a(n)^s = 2*zeta(s-1), for s > 2 (Borwein, 1994). - _Amiram Eldar_, Oct 31 2020
%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=0..100) ];
%p # More efficient:
%p a := n -> isqrt(n): seq(a(n), n=0..98); # _Peter Luschny_, Mar 13 2019
%t A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* _Enrique Pérez Herrero_, Apr 14 2010 *)
%t Flatten[Table[PadRight[{}, 2 n, n], {n, 10}]] (* _Harvey P. Dale_, Nov 16 2011 *)
%t CoefficientList[Series[x QPochhammer[-x^2, x^4] QPochhammer[x^8, x^8]/(1 - x), {x, 0, 50}], x] (* _Eric W. Weisstein_, Jan 10 2024 *)
%o (PARI) {a(n) = ceil( sqrtint(4*n) / 2)}; /* _Michael Somos_, Feb 11 2004 */
%o (PARI) a(n)=(sqrtint(4*n) + 1)\2 \\ _Charles R Greathouse IV_, Jun 08 2020
%o (Haskell)
%o a000194 n = a000194_list !! (n-1)
%o a000194_list = concat $ zipWith ($) (map replicate [2,4..]) [1..]
%o -- _Reinhard Zumkeller_, Mar 18 2011
%o (Python)
%o from math import isqrt
%o def A000194(n): return (m:=isqrt(n))+int(n-m*(m+1)>=1) # _Chai Wah Wu_, Jul 30 2022
%Y Cf. A000037, A002024, A002061, A002378, A053187, A105209, A259351.
%Y Partial sums of A005369.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, May 31 2000
%E Edited by _M. F. Hasler_, Mar 01 2014
%E Initial 0 added by _N. J. A. Sloane_, Nov 13 2017
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