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%I #114 Sep 16 2022 03:58:00
%S 1,4,7,12,17,24,31,40,49,60,71,84,97,112,127,144,161,180,199,220,241,
%T 264,287,312,337,364,391,420,449,480,511,544,577,612,647,684,721,760,
%U 799,840,881,924,967,1012,1057,1104,1151,1200,1249,1300,1351,1404,1457
%N a(n) = n + floor(n^2/2).
%C Last term in each group in A074147.
%C Index of the last occurrence of n in A100795.
%C 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
%C a(n) = A214075(n+2,2). - _Reinhard Zumkeller_, Jul 03 2012
%C 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
%C 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
%C 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
%C 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
%C Row sums of A143182 triangle - _Nikita Sadkov_, Oct 10 2018
%H Vincenzo Librandi, <a href="/A074148/b074148.txt">Table of n, a(n) for n = 1..2000</a>
%H D. Bevan, C. Homberger, and B. E. Tenner, <a href="http://arxiv.org/abs/1608.06931">Prolific permutations and permuted packings: downsets containing many large patterns</a>, arXiv preprint arXiv:1608.06931 [math.CO], 2016.
%H Peter M. Chema, <a href="/A074148/a074148_1.pdf">Illustration of initial terms, n > 1</a>.
%H A. Edelman and M. La Croix, <a href="http://arxiv.org/abs/1410.7065">The Singular Values of the GUE (Less is More)</a>, arXiv preprint arXiv:1410.7065 [math.PR], 2014-2015. See Section 7.
%H Kival Ngaokrajang, <a href="/A074148/a074148.pdf">Illustration of initial terms</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LongestPathProblem.html">Longest Path Problem</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cartan_decomposition">Cartan decomposition</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = (2*n^2 + 4*n + (-1)^n - 1)/4. - _Vladeta Jovovic_, Apr 06 2003
%F a(n) = A109225(n,2) for n > 1. - _Reinhard Zumkeller_, Jun 23 2005
%F a(n) = +2*a(n-1) - 2*a(n-3) + 1*a(n-4). - _Joerg Arndt_, Apr 02 2011
%F a(n) = a(n-2) + 2*n, a(0) = 0, a(1) = 1. - _Paul Barry_, Jul 17 2004
%F From _R. J. Mathar_, Aug 30 2008: (Start)
%F G.f.: x*(1 + 2*x - x^2)/((1 - x)^3*(1 + x)).
%F a(n) + a(n+1) = A028387(n).
%F a(n+1) - a(n) = A109613(n+1). (End)
%F a(n) = floor(n^4/(2n^2 + 1)) with offset 2..a(2) = 1. - _Gary Detlefs_, Feb 11 2010
%F a(n) = n + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 14 2013
%F From _Franck Maminirina Ramaharo_, Jan 04 2019: (Start)
%F a(n) = n*(n + 1)/2 + floor(n/2) = A000217(n) + A004526(n).
%F E.g.f.: (exp(-x) - (1 - 6*x - 2*x^2)*exp(x))/4. (End)
%F Sum_{n>=1} 1/a(n) = 1 - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - _Amiram Eldar_, Sep 16 2022
%e Equals row sums of the generating triangle:
%e 1;
%e 3, 1;
%e 5, 1, 1;
%e 7, 1, 3, 1;
%e 9, 1, 5, 1, 1;
%e 11, 1, 7, 1, 3, 1;
%e 13, 1, 9, 1, 5, 1, 1;
%e 15, 1, 11, 1, 7, 1, 3, 1;
%e ...
%e Example: a(5) = 17 = (9 + 1 + 5 + 1 + 1). - _Gary W. Adamson_, May 16 2010
%e 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
%p seq(floor(n^4/(2*n^2+1)),n=2..25); # _Gary Detlefs_, Feb 11 2010
%t 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 *)
%t Table[n + Floor[n^2/2], {n, 20}] (* _Eric W. Weisstein_, Mar 27 2018 *)
%t Table[((-1)^n + 2 n (n + 2) - 1)/4, {n, 10}] (* _Eric W. Weisstein_, Mar 27 2018 *)
%t LinearRecurrence[{2, 0, -2, 1}, {1, 4, 7, 12}, 20] (* _Eric W. Weisstein_, Mar 27 2018 *)
%t CoefficientList[Series[(-1 - 2 x + x^2)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 27 2018 *)
%o (Magma) [(2*n^2+4*n+(-1)^n-1)/4: n in [1..60]]; // _Vincenzo Librandi_, Jun 16 2011
%o (PARI) a(n)=(2*n^2+4*n-1)\/4 \\ _Charles R Greathouse IV_, Apr 17 2012
%o (GAP) List([1..60],n->n+Int(n^2/2)); # _Muniru A Asiru_, Jan 04 2019
%o (Python)
%o def A074148(n): return n + n**2//2 # _Chai Wah Wu_, Jun 07 2022
%Y Cf. A074147, A074149, A100795, A061925.
%Y a(n) = A000982(n+1) - 1.
%Y Antidiagonal sums of A237447 & A237448.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, Aug 28 2002
%E More terms from _Vladeta Jovovic_, Apr 06 2003
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 31 2007
%E Further edited by _N. J. A. Sloane_, Sep 06 2008 at the suggestion of _R. J. Mathar_
%E Description simplified by _Eric W. Weisstein_, Mar 27 2018
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