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 A000267 Integer part of square root of 4n+1. 19

%I

%S 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,

%T 11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,13,14,14,14,14,14,

%U 14,14,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,17

%N Integer part of square root of 4n+1.

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

%C 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

%C Triangle A094727 read by diagonals. - _Philippe Deléham_, Mar 21 2014

%C Partial sums of A240025; a(n) = number of quarter squares <= n. - _Reinhard Zumkeller_, Jul 05 2014

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

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

%H T. D. Noe, <a href="/A000267/b000267.txt">Table of n, a(n) for n = 0..10000</a>

%H Gal Cohensius, Urban Larsson, Reshef Meir, David Wahlstedt, <a href="https://arxiv.org/abs/1805.09368">Cumulative subtraction games</a>, arXiv:1805.09368 [math.CO], 2018.

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q723.htm">Question 723</a>, J. Ind. Math. Soc.

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

%F a(n) = 1 + a(n - floor(n^(1/2))), if n>0. - _Michael Somos_, Jul 22 2002

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

%F a(n) = |{floor(n/k): k in Z+}|. - _David W. Wilson_, May 26 2005

%F a(n) = ceiling(2*sqrt(n+1) - 1). - _Mircea Merca_, Feb 03 2012

%F a(n) = A000196(A016813(n)). - _Reinhard Zumkeller_, Dec 13 2012

%F a(n) = A070939(A227368(n+1)), conjectured. - _Antti Karttunen_, Dec 28 2013

%F a(n) = floor( sqrt(n) + sqrt(n+2) ). [_Bruno Berselli_, Jan 08 2015]

%F a(n) = floor( sqrt(4*n + k) ) where k = 1, 2, or 3. - _Michael Somos_, Mar 11 2015

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

%F a(n) = 1 + A055086(n). - _Michael Somos_, Sep 02 2017

%e From _Philippe Deléham_, Mar 21 2014: (Start)

%e Triangle A094727 begins:

%e 1;

%e 2, 3;

%e 3, 4, 5;

%e 4, 5, 6, 7;

%e 5, 6, 7, 8, 9;

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

%e 1;

%e 2;

%e 3, 3;

%e 4, 4;

%e 5, 5, 5;

%e 6, 6, 6;

%e 7, 7, 7, 7;

%e 8, 8, 8, 8;

%e 9, 9, 9, 9, 9;

%e 10, 10, 10, 10, 10; (End)

%e 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 + ...

%t Table[Floor[Sqrt[4*n + 1]], {n, 0, 100}] (* _T. D. Noe_, Jun 19 2012 *)

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

%o a000267 = a000196 . a016813 -- _Reinhard Zumkeller_, Dec 13 2012

%Y Cf. A055086, A080037, A227368.

%Y Cf. A240025, A002620.

%K nonn,easy,nice,tabf

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Michael Somos_, Jun 13 2000

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Last modified December 13 12:49 EST 2018. Contains 318086 sequences. (Running on oeis4.)