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 A004116 a(n) = floor((n^2 + 6n - 3)/4). (Formerly M2524) 9
 1, 3, 6, 9, 13, 17, 22, 27, 33, 39, 46, 53, 61, 69, 78, 87, 97, 107, 118, 129, 141, 153, 166, 179, 193, 207, 222, 237, 253, 269, 286, 303, 321, 339, 358, 377, 397, 417, 438, 459 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n)-3 is the maximal size of a regular triangulation of a prism over a regular n-gon. Solution to a postage stamp problem with 2 denominations. 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 at http://www.alkauskas.puslapiai.lt/MP3/gkw.txt and evaluating degree(denom(...)). - Stephen Crowley, Sep 18 2011 From Griffin N. Macris, Jul 19 2016: (Start) 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. For a(6)=17, four functions are excluded, because: x^2 + 2x + 1 = (x+1)^2 + 0(x+1) + 0 x^2 + 2x + 2 = (x+1)^2 + 0(x+1) + 1 x^2 + 2x + 3 = (x+1)^2 + 0(x+1) + 2 x^2 + 3x + 2 = (x+1)^2 + 1(x+1) + 0 (End) REFERENCES 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 = 1..10000 G. Alkauskas, Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator, arXiv:1004.1783 [math.NT], 2010-2012. R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210. M. Develin, Maximal triangulations of a regular prism, arXiv:math/0309220 [math.CO], 2003. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 420 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. David Singmaster, David Fielker, N. J. A. Sloane, Correspondence, August 1979 Wikipedia, Gauss-Kuzmin-Wirsing operator Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(n) = floor((1/4)*n^2 + (3/2)*n + 1/4) - 1. a(n) = (1/8)*(-1)^(n+1) - 7/8 + (3/2)*n + (1/4)*n^2. From Ilya Gutkovskiy, Jul 20 2016: (Start) O.g.f.: x*(1 + x - x^3)/((1 - x)^3*(1 + x)). E.g.f.: (8 + sinh(x) - cosh(x) + (2*x^2 + 14*x - 7)*exp(x))/8. a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). a(n) = Sum_{k=0..n-1} A266977(k). (End) MAPLE A004116:=(-1-z+z**3)/(z+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA Table[Floor[(n^2 + 6 n - 3)/4], {n, 40}] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 9}, 40] (* Michael De Vlieger, Jul 19 2016 *) PROG (PARI) a(n)=(n^2+6*n-3)>>2 (MAGMA) [Floor( (n^2 + 6*n - 3)/4 ) : n in [1..50]]; // Vincenzo Librandi, Sep 19 2011 CROSSREFS Sequence in context: A109512 A025205 A024190 * A004129 A219646 A185173 Adjacent sequences:  A004113 A004114 A004115 * A004117 A004118 A004119 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 14 12:02 EDT 2019. Contains 328004 sequences. (Running on oeis4.)