OFFSET
0,4
COMMENTS
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
From Wolfdieter Lang, Mar 12 2019: (Start)
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).
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).
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)
REFERENCES
Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 59-60.
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.
G. Gutin, Problem 913 (BCC20.5), Mediated digraphs, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
a(n) = A000037(n) - n.
G.f.: x * f(x^2, x^6)/(1-x) where f(,) is Ramanujan's two-variable theta function. - Michael Somos, May 31 2000
a(n) = a(n - 2*a(n - a(n-1))) + 1. - Benoit Cloitre, Oct 27 2002
a(n+1) = a(n) + A005369(n).
a(n) = floor((1/2)*(1 + sqrt(4*n - 3))). - Zak Seidov, Jan 18 2006
a(n) = A000037(n) - n. - Jaroslav Krizek, Jun 14 2009
a(n) = floor(A027434(n)/2). - Gregory R. Bryant, Apr 17 2013
From Mikael Aaltonen, Jan 17 2015: (Start)
a(n) = floor(sqrt(n) + 1/2).
a(n) = sqrt(A053187(n)). (End)
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
a(n) = floor(2*sqrt(n)) - floor(sqrt(n)). - Ridouane Oudra, Jun 08 2020
Sum_{n>=1} 1/a(n)^s = 2*zeta(s-1), for s > 2 (Borwein, 1994). - Amiram Eldar, Oct 31 2020
EXAMPLE
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 + ...
MAPLE
Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=0..100) ];
# More efficient:
a := n -> isqrt(n): seq(a(n), n=0..98); # Peter Luschny, Mar 13 2019
MATHEMATICA
A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* Enrique Pérez Herrero, Apr 14 2010 *)
Flatten[Table[PadRight[{}, 2 n, n], {n, 10}]] (* Harvey P. Dale, Nov 16 2011 *)
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 *)
PROG
(PARI) {a(n) = ceil( sqrtint(4*n) / 2)}; /* Michael Somos, Feb 11 2004 */
(PARI) a(n)=(sqrtint(4*n) + 1)\2 \\ Charles R Greathouse IV, Jun 08 2020
(PARI) apply( {A000194(n)=sqrtint(4*n)\/2}, [0..99]) \\ M. F. Hasler, Jun 22 2024
(Haskell)
a000194 n = a000194_list !! (n-1)
a000194_list = concat $ zipWith ($) (map replicate [2, 4..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011
(Python)
from math import isqrt
def A000194(n): return (m:=isqrt(n))+int(n-m*(m+1)>=1) # Chai Wah Wu, Jul 30 2022
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, May 31 2000
Edited by M. F. Hasler, Mar 01 2014
Initial 0 added by N. J. A. Sloane, Nov 13 2017
STATUS
approved