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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 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. See also code.
• 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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• David Singmaster, David Fielker, and 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).

Programs

Magma

[Floor( (n^2 + 6*n - 3)/4 ) : n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

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 *)

PARI

a(n)=(n^2+6*n-3)>>2

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)

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