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A000267 Integer part of square root of 4n+1. 19
1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

1^1, 2^1, 3^2, 4^2, 5^3, 6^3, 7^4, 8^4, 9^5, 10^5, ...

Start with n, repeatedly subtract the square root of the previous term; a(n) gives number of steps to reach 0. - Robert G. Wilson v, Jul 22 2002

Triangle A094727 read by diagonals. - Philippe Deléham, Mar 21 2014

Partial sums of A240025; a(n) = number of quarter squares <= n. -- Reinhard Zumkeller, Jul 05 2014

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 20.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 77, Entry 23.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

S. Ramanujan, Question 723, J. Ind. Math. Soc.

FORMULA

floor(a(n)/2) = A000196(n).

a(n) = 1 + a(n - floor(n^(1/2))), if n>0. - Michael Somos, Jul 22 2002

a(n) = floor( 1 / ( sqrt(n + 1) - sqrt(n) ) ). - Robert A. Stump (bob_ess107(AT)yahoo.com), Apr 07 2003

a(n) = |{floor(n/k): k in Z+}|. - David W. Wilson, May 26 2005

a(n) = ceiling(2*sqrt(n+1) - 1). - Mircea Merca, Feb 03 2012

a(n) = A000196(A016813(n)). - Reinhard Zumkeller, Dec 13 2012

a(n) = A070939(A227368(n+1)), conjectured. - Antti Karttunen, Dec 28 2013

a(n) = floor( sqrt(n) + sqrt(n+2) ). [Bruno Berselli, Jan 08 2015]

a(n) = floor( sqrt(4*n + k) ) where k = 1, 2, or 3. - Michael Somos, Mar 11 2015

G.f.: (Sum_{k>0} x^floor(k^2 / 4)) / (1 - x). - Michael Somos, Mar 11 2015

a(n) = 1 + A055086(n). - Michael Somos, Sep 02 2017

EXAMPLE

Triangle A094727 begins:

1;

2, 3;

3, 4, 5;

4, 5, 6, 7;

5, 6, 7, 8, 9;

6, 7, 8, 9, 10, 11; ...

Read by diagonals:

1;

2;

3,   3;

4,   4;

5,   5,  5;

6,   6,  6;

7,   7,  7,  7;

8,   8,  8,  8;

9,   9,  9,  9,  9;

10, 10, 10, 10, 10; - Philippe Deléham, Mar 21 2014

G.f. = 1 + 2*x + 3*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 5*x^7 + 5*x^8 + 6*x^9 + ...

MATHEMATICA

Table[Floor[Sqrt[4*n + 1]], {n, 0, 100}] (* T. D. Noe, Jun 19 2012 *)

PROG

(PARI) {a(n) = if( n<0, 0, sqrtint(4*n + 1))};

(Haskell)

a000267 = a000196 . a016813  -- Reinhard Zumkeller, Dec 13 2012

CROSSREFS

Cf. A055086, A080037, A227368.

Cf. A240025, A002620.

Sequence in context: A279783 A132663 A023964 * A249728 A060020 A166127

Adjacent sequences:  A000264 A000265 A000266 * A000268 A000269 A000270

KEYWORD

nonn,easy,nice,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos, Jun 13 2000

STATUS

approved

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Last modified December 13 02:44 EST 2017. Contains 295954 sequences.