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 A000982 a(n) = ceiling(n^2/2). (Formerly M1348 N0517) 98
 0, 1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also, floor( (n^2+1)/2 ). - N. J. A. Sloane, Feb 08 2019 Floor(arithmetic mean of next n numbers). - Amarnath Murthy, Mar 11 2003 Pairwise sums of repeated squares (A008794). Also, number of topologies on n+1 unlabeled elements with exactly 4 elements in the topology. a(3) gives 4 elements a,b,c,d; the valid topologies are (0,a,ab,abcd), (0,a,abc,abcd), (0,ab,abc,abcd), (0,a,bcd,abcd) and (0,ab,cd,abcd), with a count of 5. - Jon Perry, Mar 05 2004 Partition n in two parts, say r and s so that r^2 + s^2 is minimal, then a(n) = r^2 +s^2. Geometrical significance: folding a rod with length n units at right angles in such a way that the end points are at the least distance, which is given by a(n)^(1/2) as the hypotenuse of a right triangle with the sum of the base and height = n units. - Amarnath Murthy, Apr 18 2004 Convolution of A002061(n)-0^n and (-1)^n. Convolution of n (A001477) with {1,0,2,0,2,0,2,...}. Partial sums of repeated odd numbers {0,1,1,3,3,5,5,...}. - Paul Barry, Jul 22 2004 The ratio of the sum of terms over the total number of terms in an n X n spiral. The sum of terms of an n X n spiral is A037270, or Sum_{k=0..n^2} k = (n^4 + n^2)/2 and the total number of terms is n^2. - William A. Tedeschi, Feb 27 2008 Starting with offset 1 = row sums of triangle A158946. - Gary W. Adamson, Mar 31 2009 Partial sums of A109613. - Reinhard Zumkeller, Dec 05 2009 Also the number of compositions of even natural numbers into 2 parts < n. For example a(3)=5 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2) of even natural numbers into 2 parts < 3. a(4)=8 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2), (1,3), (3,1), (3,3) of even natural numbers into 2 parts < 4. - Adi Dani, Jun 05 2011 A001105 and A001844 interleaved. - Omar E. Pol, Sep 18 2011 Number of (w,x,y) having all terms in {0,...,n} and w=average(x,y). - Clark Kimberling, May 15 2012 For n > 0, minimum number of lines necessary to get through all unit cubes of an n X n X n cube (see Kantor link). - Michel Marcus, Apr 13 2013 Sum_{n > 0} 1/a(n) = Sum_{n > 0} 1/(2*n^2) + Sum_{n >= 0} 1/(2*n + 2*n^2 + 1) = (zeta(2) + (Pi* tanh(Pi/2)))/2 = 2.26312655.... - Enrique Pérez Herrero, Jun 17 2013 For n > 1, a(n) is the edge cover number of the n X n king graph. - Eric W. Weisstein, Jun 20 2017 Also the number of vertices in the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017 The same sequence arises in the triangular array of the integers >= 1, according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array, and the second row of that sub-array (with apex a(n-1)) contains just two numbers, one odd, one even. The one with opposite parity to a(n-1) is a(n). - David James Sycamore, Jul 29 2018 Size of minimal ternary 1-covering code with code length n, i.e., K_n(3,1). See Kalbfleisch and Stanton. - Patrick Wienhöft, Jan 29 2019 For n > 1, a(n-1) is the maximum number of inversions in a permutation consisting of a single n-cycle on n symbols. - M. Ryan Julian Jr., Sep 10 2019 Also the number of classes of convex inscribed polyominoes in a (2,n) rectangular grid; two polyominoes are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other. - Jean-Luc Manguin, Jan 29 2020 a(n) is the number of pairs (p,q) such that 1 <= p, p+1 < q <= n+2 and q <> 2*p. - César Eliud Lozada, Oct 25 2020 a(n) is the maximum number of copies of a 12 permutation pattern in an alternating (or zig-zag) permutation of length n+1. The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous. - Lara Pudwell, Dec 01 2020 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..3000 M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,4) in theorem 5. J. G. Kalbfleisch and R. G. Stanton, A combinatorial problem in matching, J. London Math. Soc. Vol. 1, No. 1 (1969), 60-64. [Corrected by N. J. A. Sloane, Feb 08 2019] J. M. Kantor, Mathématiques venues d'ailleurs: divertissements mathématiques en U.R.S.S., Le cube transpercé, pp. 56-62, Belin, Paris, 1982. S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Eric Weisstein's World of Mathematics, Black Bishop Graph Eric Weisstein's World of Mathematics, Edge Cover Number Eric Weisstein's World of Mathematics, King Graph Eric Weisstein's World of Mathematics, Topology Eric Weisstein's World of Mathematics, Vertex Count Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(2*n) = 2*n^2, a(2*n+1) = 2*n^2 + 2*n + 1. G.f.: -x*(1+x^2) / ( (1+x)*(x-1)^3 ). - Simon Plouffe in his 1992 dissertation From Benoit Cloitre, Nov 06 2002: (Start) a(n) = (2*n^2 + 1 - (-1)^n) / 4. a(0)=0, a(1)=1; for n>1, a(n+1) = n + 1 + max(2*floor(a(n)/2), 3*floor(a(n)/3)). (End) G.f.: (x + x^2 + x^3 + x^4)/((1 - x)*(1 - x^2)^2), not reduced. - Len Smiley a(n) = a(n-2) + 2n - 2. - Paul Barry, Jul 17 2004 From Paul Barry, Jul 22 2004: (Start) G.f.: x*(1+x^2)/((1-x^2)*(1-x)^2) = x*(1+x^2)/((1+x)*(1-x)^3); a(n) = Sum_{k=0..n} (k^2 - k + 1 - 0^k)*(-1)^(n-k); a(n) = Sum_{k=0..n} (1 + (-1)^(n-k) - 0^(n-k))*k. (End) From Reinhard Zumkeller, Feb 27 2006: (Start) a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1. a(n) = A116940(n) - A005843(n). (End) Starting with offset 1, = row sums of triangle A134444. Also, with offset 1, = binomial transform of [1, 1, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007 a(n) = floor((n^2+1)/2). - William A. Tedeschi, Feb 27 2008 a(n) = A004526(n+1) + A000217(n-1). - Yosu Yurramendi, Sep 12 2008, corrected by Klaus Purath, Jun 15 2021 From Jaume Oliver Lafont, Dec 05 2008: (Start) a(n) = a(n-1) + a(n-2) - a(n-3) + 2. a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End) a(n) = A004526(n)^2 + A110654(n)^2. - Philippe Deléham, Mar 12 2009 a(n) = n^2 - floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013 Euler transform is length 4 sequence [2, 2, 0, -1]. a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015 a(n) is also the number of independent entries in a centrosymmetric n X n matrix: M(i, j) = M(n-i+1, n-j+1). - Wolfdieter Lang, Oct 12 2015 For n > 1, a(n+1)/a(n) = 3 - A081352(n-2)/a(n). - Miko Labalan, Mar 26 2016 E.g.f.: (1/2)*(x*(1 + x)*cosh(x) + (1 + x + x^2)*sinh(x)). - Stefano Spezia, Feb 03 2020 a(n) = binomial(n+1,2) - floor(n/2). - César Eliud Lozada, Oct 25 2020 From Klaus Purath, Jun 15 2021: (Start) a(n-1) + a(n) = A002061(n). a(n) = (a(n-1)^2 + 1) / a(n-2), n >= 3 odd. a(n) = (a(n-1)^2 - (n-1)^2) / a(n-2), n >= 4 even. (End) EXAMPLE G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 13*x^5 + 18*x^6 + 25*x^7 + 32*x^8 + ... Centrosymmetric 3 X 3 matrix: [[a,b,c],[d,e,d],[c,b,a]], a(3) = 3*(3-1)/2 + (3-1)/2 + 1 = (3^2+1)/2  = 5 from a,b,c,d,e. 4 X 4 case: [[a,b,c,d],[e,f,g,h],[h,g,f,e],[d,c,b,a]], a(4) = 4*4/2 = 8. - Wolfdieter Lang, Oct 12 2015 a(3) = 5. The alternating permutation of length 3 + 1 = 4 with the maximum number of copies of 123 is 1324. The five copies are 12, 13, 14, 23, and 24. - Lara Pudwell, Dec 01 2020 MAPLE seq( ceil(n^2/2), n=0..30) ; # R. J. Mathar, Jun 05 2011 MATHEMATICA Table[Ceiling[n^2/2], {n, 0, 120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *) Accumulate[Join[{0}, (# - Boole[EvenQ[#]] &) /@ Range]] (* Alonso del Arte, Sep 11 2019 *) PROG (Magma) [(2*n^2 + 1 - (-1)^n) / 4: n in [0..60]]; // Vincenzo Librandi, Jun 16 2011 (Haskell) a000982 = (`div` 2) . (+ 1) . (^ 2)  -- Reinhard Zumkeller, Jun 27 2013 (PARI) a(n)=(n^2+1)\2 \\ Charles R Greathouse IV, Sep 13 2013 (PARI) x='x+O('x^100); concat(, Vec(x*(1+x^2)/((1+x)*(1-x)^3))) \\ Altug Alkan, Oct 12 2015 (PARI) apply( A000982(n)=n^2\/2, [0..55]) \\ M. F. Hasler, Feb 29 2020 (Scala) (((1 to 49) by 2) flatMap { List.fill(2)(_) }).scanLeft(0)(_ + _) // Alonso del Arte, Sep 11 2019 CROSSREFS Cf. A000096, A134444, A037270, A158946, A001844, A001105, A007590, A168380. Column 2 of A195040. Sequence in context: A049617 A054074 A290268 * A289751 A200274 A122221 Adjacent sequences:  A000979 A000980 A000981 * A000983 A000984 A000985 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified September 28 01:47 EDT 2022. Contains 357063 sequences. (Running on oeis4.)