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A000194 n appears 2n times; also nearest integer to square root of n. 29
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) = inverse (frequency distribution) sequence of A002378(n-1). - Jaroslav Krizek, Jun 14 2009

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

REFERENCES

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 = 1..1000

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.

M. Somos, Sequences used for indexing triangular or square arrays

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

G.f.: f(x^2, x^6) * x / (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)

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=1..100) ];

MATHEMATICA

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* Enrique Pérez Herrero, Apr 14 2010 *)

Flatten[Table[PadRight[{}, 2n, n], {n, 10}]] (* Harvey P. Dale, Nov 16 2011 *)

PROG

(PARI) {a(n) = ceil( sqrtint(4*n) / 2)}; /* Michael Somos, Feb 11 2004 */

(Haskell)

a000194 n = a000194_list !! (n-1)

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

-- Reinhard Zumkeller, Mar 18 2011

CROSSREFS

Partial sums of A005369.

A000037(n) - n.

Cf. A002024, A259351, A002378.

Sequence in context: A260999 A090532 A003058 * A168255 A097429 A100617

Adjacent sequences:  A000191 A000192 A000193 * A000195 A000196 A000197

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Michael Somos, May 31 2000

Edited by M. F. Hasler, Mar 01 2014

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.