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A004116 a(n) = floor((n^2 + 6n - 3)/4).

%I A004116 M2524 #74 Aug 13 2022 06:23:33
%S A004116 1,3,6,9,13,17,22,27,33,39,46,53,61,69,78,87,97,107,118,129,141,153,
%T A004116 166,179,193,207,222,237,253,269,286,303,321,339,358,377,397,417,438,
%U A004116 459
%N A004116 a(n) = floor((n^2 + 6n - 3)/4).
%C A004116 a(n)-3 is the maximal size of a regular triangulation of a prism over a regular n-gon.
%C A004116 Solution to a postage stamp problem with 2 denominations.
%C A004116 This sequence is half the degree of the denominator of a certain sequence of rational polynomials defined in the referenced paper by G. Alkauskas. Although this fact is not documented in the paper it can be verified by running the author's code and evaluating degree(denom(...)). - _Stephen Crowley_, Sep 18 2011
%C A004116 From _Griffin N. Macris_, Jul 19 2016: (Start)
%C A004116 Consider quadratic functions x^2+ax+b. Then a(n) is the number of these functions with 0 <= a+b < n, modulo changing x to x+c for a constant c.
%C A004116 For a(6)=17, four functions are excluded, because:
%C A004116 x^2 + 2x + 1 = (x+1)^2 + 0(x+1) + 0
%C A004116 x^2 + 2x + 2 = (x+1)^2 + 0(x+1) + 1
%C A004116 x^2 + 2x + 3 = (x+1)^2 + 0(x+1) + 2
%C A004116 x^2 + 3x + 2 = (x+1)^2 + 1(x+1) + 0 (End)
%D A004116 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004116 Vincenzo Librandi, <a href="/A004116/b004116.txt">Table of n, a(n) for n = 1..10000</a>
%H A004116 G. Alkauskas, <a href="http://arxiv.org/abs/1004.1783">Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator</a>, arXiv:1004.1783 [math.NT], 2010-2012. See also <a href="https://web.vu.lt/mif/g.alkauskas/MP3/gkw.txt">code</a>.
%H A004116 R. Alter and J. A. Barnett, <a href="http://www.jstor.org/stable/2321610">A postage stamp problem</a>, Amer. Math. Monthly, 87 (1980), 206-210.
%H A004116 M. Develin, <a href="http://arXiv.org/abs/math.CO/0309220">Maximal triangulations of a regular prism</a>, arXiv:math/0309220 [math.CO], 2003.
%H A004116 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=420">Encyclopedia of Combinatorial Structures 420</a>.
%H A004116 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A004116 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A004116 David Singmaster, David Fielker, and N. J. A. Sloane, <a href="/A004116/a004116.pdf">Correspondence, August 1979</a>.
%H A004116 Wikipedia, <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin%E2%80%93Wirsing_operator">Gauss-Kuzmin-Wirsing operator</a>.
%H A004116 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A004116 a(n) = floor((1/4)*n^2 + (3/2)*n + 1/4) - 1.
%F A004116 a(n) = (1/8)*(-1)^(n+1) - 7/8 + (3/2)*n + (1/4)*n^2.
%F A004116 From _Ilya Gutkovskiy_, Jul 20 2016: (Start)
%F A004116 O.g.f.: x*(1 + x - x^3)/((1 - x)^3*(1 + x)).
%F A004116 E.g.f.: (8 + sinh(x) - cosh(x) + (2*x^2 + 14*x - 7)*exp(x))/8.
%F A004116 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F A004116 a(n) = Sum_{k=0..n-1} A266977(k). (End)
%F A004116 Sum_{n>=1} 1/a(n) = 2 + tan(sqrt(13)*Pi/2)*Pi/sqrt(13) - cot(sqrt(3)*Pi)*Pi/(2*sqrt(3)). - _Amiram Eldar_, Aug 13 2022
%p A004116 A004116:=(-1-z+z**3)/(z+1)/(z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A004116 Table[Floor[(n^2 + 6 n - 3)/4], {n, 40}] (* or *)
%t A004116 LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 9}, 40] (* _Michael De Vlieger_, Jul 19 2016 *)
%o A004116 (PARI) a(n)=(n^2+6*n-3)>>2
%o A004116 (Magma) [Floor( (n^2 + 6*n - 3)/4 ) : n in [1..50]]; // _Vincenzo Librandi_, Sep 19 2011
%K A004116 nonn,easy
%O A004116 1,2
%A A004116 _N. J. A. Sloane_