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 A110654 a(n) = ceiling(n/2), or: a(2*k) = k, a(2*k+1) = k+1. 54
 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, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The number of partitions of 2n into exactly 2 odd parts. - Wesley Ivan Hurt, Jun 01 2013 Number of nonisomorphic outer planar graphs of order n >= 3 and size n+1. - Christian Barrientos and Sarah Minion, Feb 27 2018 Also the clique covering number of the n-dipyramidal graph for n >= 3. - Eric W. Weisstein, Jun 27 2018 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Clique Covering Number Eric Weisstein's World of Mathematics, Dipyramidal Graph Wikipedia, The Free Encyclopedia Floor and ceiling functions Index entries for linear recurrences with constant coefficients, signature (1,1,-1). FORMULA a(n) = floor(n/2) + n mod 2. a(n) = A004526(n+1) = A001057(n)*(-1)^(n+1). For n > 0: a(n) = A008619(n-1). A110655(n) = a(a(n)), A110656(n) = a(a(a(n))). a(n) = A109613(n) - A028242(n) = A110660(n) / A028242(n). a(n) = A001222(A029744(n)). - Reinhard Zumkeller, Feb 16 2006 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 First differences of quarter-squares: a(n) = A002620(n+1) - A002620(n). - Reinhard Zumkeller, Aug 06 2009 a(n) = A007742(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010 a(n) = A000217(n) / A008619(n). - Reinhard Zumkeller, Aug 24 2011 From Michael Somos, Sep 19 2006: (Start) Euler transform of length 2 sequence [1, 1]. G.f.: x/((1-x)*(1-x^2)). a(-1-n) = -a(n). (End) 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 a(n) = A065033(n) for n > 0. - R. J. Mathar, Aug 18 2008 a(n) = 1/4 - (-1)^n/4 + n/2. - Paolo P. Lava, Oct 03 2008 a(n) = ceiling(n/2) = smallest integer >= n/2. - M. F. Hasler, Nov 17 2008 If n is zero then a(n) is zero, else a(n) = a(n-1) + (n mod 2). - R. J. Cano, Jun 15 2014 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 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 a(n) = (n + (n mod 2)) / 2. - Fred Daniel Kline, Jun 08 2016 E.g.f.: (sinh(x) + x*exp(x))/2. - Ilya Gutkovskiy, Jun 08 2016 EXAMPLE 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 + ... MAPLE a:=n->add(chrem( [n, j], [1, 2] ), j=1..n):seq(a(n), n=0..75); # Zerinvary Lajos, Apr 08 2009 MATHEMATICA a[ n_] := Ceiling[ n / 2]; (* Michael Somos, Jun 15 2014 *) a[ n_] := Quotient[ n, 2, -1]; (* Michael Somos, Jun 15 2014 *) a = 0; a[n_] := a[n] = n - a[n - 1]; Table[a[n], {n, 0, 100}] (* Carlos Eduardo Olivieri, Dec 22 2014 *) CoefficientList[Series[x^/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *) LinearRecurrence[{1, 1, -1}, {0, 1, 1}, 75] (* Robert G. Wilson v, Feb 05 2015 *) PROG (PARI) a(n)=n\2+n%2; (PARI) a(n)=(n+1)\2; \\ M. F. Hasler (Sage) [floor(n/2) + 1 for n in xrange(-1, 75)] # Zerinvary Lajos, Dec 01 2009 (Haskell) a110654 = (`div` 2) . (+ 1) a110654_list = tail a004526_list  -- Reinhard Zumkeller, Jul 27 2012 (MAGMA) [Ceiling(n/2): n in [0..80]]; // Vincenzo Librandi, Nov 04 2014 CROSSREFS Essentially the same sequence as A008619 and A123108. Cf. A014557. Cf. A298648 (number of smallest coverings of dipyramidal graphs by maximal cliques). Sequence in context: A140106 A123108 A008619 * A109728 A157271 A025162 Adjacent sequences:  A110651 A110652 A110653 * A110655 A110656 A110657 KEYWORD nonn,easy AUTHOR Reinhard Zumkeller, Aug 05 2005 EXTENSIONS Deleted wrong formula, added formula and better PARI code. - M. F. Hasler, Nov 17 2008 STATUS approved

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Last modified May 19 14:58 EDT 2019. Contains 323395 sequences. (Running on oeis4.)