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 A074148 a(n) = n + floor(n^2/2). 41
 1, 4, 7, 12, 17, 24, 31, 40, 49, 60, 71, 84, 97, 112, 127, 144, 161, 180, 199, 220, 241, 264, 287, 312, 337, 364, 391, 420, 449, 480, 511, 544, 577, 612, 647, 684, 721, 760, 799, 840, 881, 924, 967, 1012, 1057, 1104, 1151, 1200, 1249, 1300, 1351, 1404, 1457 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Last term in each group in A074147. Index of the last occurrence of n in A100795. Equals row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 1, 1, ...). - Gary W. Adamson, May 16 2010 a(n) = A214075(n+2,2). - Reinhard Zumkeller, Jul 03 2012 The heart pattern appears in (n+1) X (n+1) coins. Abnormal orientation heart is A065423. Normal heart is A093005 (A074148 - A065423). Void is A007590. See illustration in links. - Kival Ngaokrajang, Sep 11 2013 a(n+1) is the smallest size of an n-prolific permutation; a permutation of s letters is n-prolific if each (s - n)-subset of the letters in its one-line notation forms a unique pattern. - David Bevan, Nov 30 2016 For n > 2, a(n-1) is the smallest size of a nontrivial permuted packing of diamond tiles with diagonal length n; a permuted packing is a translational packing for which the set of translations is the plot of a permutation. - David Bevan, Nov 30 2016 Also the length of a longest path in the (n+1) X (n+1) bishop and black bishop graphs. - Eric W. Weisstein, Mar 27 2018 Row sums of A143182 triangle - Nikita Sadkov, Oct 10 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..2000 D. Bevan, C. Homberger, and B. E. Tenner, Prolific permutations and permuted packings: downsets containing many large patterns, arXiv preprint arXiv:1608.06931 [math.CO], 2016. Peter M. Chema, Illustration of initial terms, n > 1 A. Edelman, M. La Croix, The Singular Values of the GUE (Less is More), arXiv preprint arXiv:1410.7065 [math.PR], 2014-2015. See Section 7. Kival Ngaokrajang, Illustration of initial terms Eric Weisstein's World of Mathematics, Bishop Graph Eric Weisstein's World of Mathematics, Black Bishop Graph Eric Weisstein's World of Mathematics, Longest Path Problem Wikipedia, Cartan decomposition Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(n) = (2*n^2 + 4*n + (-1)^n - 1)/4. - Vladeta Jovovic, Apr 06 2003 a(n) = A109225(n,2) for n > 1. - Reinhard Zumkeller, Jun 23 2005 a(n) = +2*a(n-1) - 2*a(n-3) + 1*a(n-4). - Joerg Arndt, Apr 02 2011 a(n) = a(n-2) + 2*n, a(0) = 0, a(1) = 1. - Paul Barry, Jul 17 2004 From R. J. Mathar, Aug 30 2008: (Start) G.f.: x*(1 + 2*x - x^2)/((1 - x)^3*(1 + x)). a(n) + a(n+1) = A028387(n). a(n+1) - a(n) = A109613(n+1). (End) a(n) = floor(n^4/(2n^2 + 1)) with offset 2..a(2) = 1. - Gary Detlefs, Feb 11 2010 a(n) = n + floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013 From Franck Maminirina Ramaharo, Jan 04 2019: (Start) a(n) = n*(n + 1)/2 + floor(n/2) = A000217(n) + A004526(n). E.g.f.: (exp(-x) - (1 - 6*x - 2*x^2)*exp(x))/4. (End) EXAMPLE Equals row sums of the generating triangle:    1;    3,  1;    5,  1,  1;    7,  1,  3,  1;    9,  1,  5,  1,  1;   11,  1,  7,  1,  3,  1;   13,  1,  9,  1,  5,  1,  1;   15,  1, 11,  1,  7,  1,  3,  1;   ... Example: a(5) = 17 = (9 + 1 + 5 + 1 + 1). - Gary W. Adamson, May 16 2010 The smallest 1-prolific permutations are 3142 and its symmetries; a(2) = 4. The smallest 2-prolific permutations are 3614725 and its symmetries; a(3) = 7. - David Bevan, Nov 30 2016 MAPLE seq(floor(n^4/(2*n^2+1)), n=2..25); # Gary Detlefs, Feb 11 2010 MATHEMATICA f[x_, y_] := Floor[Abs[y/x - x/y]]; Table[Floor[f[1, n^2 + 2 n + 1]/2], {n, 60}] (* Robert G. Wilson v, Aug 11 2010 *) Table[n + Floor[n^2/2], {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *) Table[((-1)^n + 2 n (n + 2) - 1)/4, {n, 10}] (* Eric W. Weisstein, Mar 27 2018 *) LinearRecurrence[{2, 0, -2, 1}, {1, 4, 7, 12}, 20] (* Eric W. Weisstein, Mar 27 2018 *) CoefficientList[Series[(-1 - 2 x + x^2)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *) PROG (MAGMA) [(2*n^2+4*n+(-1)^n-1)/4: n in [1..60]]; // Vincenzo Librandi, Jun 16 2011 (PARI) a(n)=(2*n^2+4*n-1)\/4 \\ Charles R Greathouse IV, Apr 17 2012 (GAP) List([1..60], n->n+Int(n^2/2)); # Muniru A Asiru, Jan 04 2019 CROSSREFS Cf. A074147, A074149, A100795, A061925. a(n) = A000982(n+1) - 1. Antidiagonal sums of A237447 & A237448. Sequence in context: A310791 A246399 A276222 * A336660 A310792 A178907 Adjacent sequences:  A074145 A074146 A074147 * A074149 A074150 A074151 KEYWORD nonn,easy AUTHOR Amarnath Murthy, Aug 28 2002 EXTENSIONS More terms from Vladeta Jovovic, Apr 06 2003 Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007 Further edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar Description simplified by Eric W. Weisstein, Mar 27 2018 STATUS approved

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Last modified April 23 05:42 EDT 2021. Contains 343199 sequences. (Running on oeis4.)