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%I #124 Feb 08 2024 01:41:48
%S 0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,
%T 15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,
%U 26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38
%N a(n) = ceiling(n/2), or: a(2*k) = k, a(2*k+1) = k+1.
%C The number of partitions of 2n into exactly 2 odd parts. - _Wesley Ivan Hurt_, Jun 01 2013
%C Number of nonisomorphic outer planar graphs of order n >= 3 and size n+1. - _Christian Barrientos_ and _Sarah Minion_, Feb 27 2018
%C Also the clique covering number of the n-dipyramidal graph for n >= 3. - _Eric W. Weisstein_, Jun 27 2018
%H Charles R Greathouse IV, <a href="/A110654/b110654.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal Graph</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions">Floor and ceiling functions</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = floor(n/2) + n mod 2.
%F a(n) = A004526(n+1) = A001057(n)*(-1)^(n+1).
%F For n > 0: a(n) = A008619(n-1).
%F A110655(n) = a(a(n)), A110656(n) = a(a(a(n))).
%F a(n) = A109613(n) - A028242(n) = A110660(n) / A028242(n).
%F a(n) = A001222(A029744(n)). - _Reinhard Zumkeller_, Feb 16 2006
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2, a(2) = a(1) = 1, a(0) = 0. - _Reinhard Zumkeller_, May 22 2006
%F First differences of quarter-squares: a(n) = A002620(n+1) - A002620(n). - _Reinhard Zumkeller_, Aug 06 2009
%F a(n) = A007742(n) - A173511(n). - _Reinhard Zumkeller_, Feb 20 2010
%F a(n) = A000217(n) / A008619(n). - _Reinhard Zumkeller_, Aug 24 2011
%F From _Michael Somos_, Sep 19 2006: (Start)
%F Euler transform of length 2 sequence [1, 1].
%F G.f.: x/((1-x)*(1-x^2)).
%F a(-1-n) = -a(n). (End)
%F a(n) = floor((n+1)/2) = |Sum_{m=1..n} Sum_{k=1..m} (-1)^k|, where |x| is the absolute value of x. - _William A. Tedeschi_, Mar 21 2008
%F a(n) = A065033(n) for n > 0. - _R. J. Mathar_, Aug 18 2008
%F a(n) = ceiling(n/2) = smallest integer >= n/2. - _M. F. Hasler_, Nov 17 2008
%F If n is zero then a(n) is zero, else a(n) = a(n-1) + (n mod 2). - _R. J. Cano_, Jun 15 2014
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + x) * u * v - (u^2 - v) / 2. - _Michael Somos_, Jun 15 2014
%F Given g.f. A(x) then 2 * x^3 * (1 + x) * A(x) * A(x^2) is the g.f. of A014557. - _Michael Somos_, Jun 15 2014
%F a(n) = (n + (n mod 2)) / 2. - _Fred Daniel Kline_, Jun 08 2016
%F E.g.f.: (sinh(x) + x*exp(x))/2. - _Ilya Gutkovskiy_, Jun 08 2016
%F Satisfies the nested recurrence a(n) = a(a(n-2)) + a(n-a(n-1)) with a(1) = a(2) = 1. Cf. A004001. - _Peter Bala_, Aug 30 2022
%e G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 5*x^9 + ...
%t a[ n_] := Ceiling[ n / 2]; (* _Michael Somos_, Jun 15 2014 *)
%t a[ n_] := Quotient[ n, 2, -1]; (* _Michael Somos_, Jun 15 2014 *)
%t a[0] = 0; a[n_] := a[n] = n - a[n - 1]; Table[a[n], {n, 0, 100}] (* _Carlos Eduardo Olivieri_, Dec 22 2014 *)
%t CoefficientList[Series[x^/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* _Robert G. Wilson v_, Feb 05 2015 *)
%t LinearRecurrence[{1, 1, -1}, {0, 1, 1}, 75] (* _Robert G. Wilson v_, Feb 05 2015 *)
%o (PARI) a(n)=n\2+n%2;
%o (PARI) a(n)=(n+1)\2; \\ _M. F. Hasler_, Nov 17 2008
%o (Sage) [floor(n/2) + 1 for n in range(-1,75)] # _Zerinvary Lajos_, Dec 01 2009
%o (Haskell)
%o a110654 = (`div` 2) . (+ 1)
%o a110654_list = tail a004526_list -- _Reinhard Zumkeller_, Jul 27 2012
%o (Magma) [Ceiling(n/2): n in [0..80]]; // _Vincenzo Librandi_, Nov 04 2014
%Y Essentially the same sequence as A008619 and A123108.
%Y Cf. A014557, A275416 (multisets).
%Y Cf. A298648 (number of smallest coverings of dipyramidal graphs by maximal cliques).
%K nonn,easy
%O 0,4
%A _Reinhard Zumkeller_, Aug 05 2005
%E Deleted wrong formula and added formula. - _M. F. Hasler_, Nov 17 2008
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