OFFSET
0,3
COMMENTS
This covers squares of all consecutively increasing integers with the exception of 2.
It is actually possible to cover all nonnegative integers by using the given formula starting with n=-2, thus giving terms 2, -2, 3, 1, 8, 8, 17, 19, 30, etc. - Vladimir Joseph Stephan Orlovsky, Feb 12 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = (n+2)*(n+1)/2 + 2*(-1)^n for n>0.
From R. J. Mathar, Dec 12 2007: (Start)
O.g.f.: x*(1 +6*x -8*x^2 +3*x^3)/((1-x)^3*(1+x)) = -3 +1/(1-x)^3 + 2/(1+x).
a(n) = A000217(n+1) + 2*(-1)^n if n>0.
(End)
E.g.f.: -3 + 2*exp(-x) + (1/2)*(2 + 4*x + x^2)*exp(x). - G. C. Greubel, Oct 12 2016
From Colin Barker, Oct 13 2016: (Start)
a(n) = (-4*(-1)^n+n+n^2)/2 for n>1.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4.
(End)
EXAMPLE
a(1) = 1 because 0 + 1 = 1^2.
a(2) = 8 because 1 + 8 = 9 = 3^2.
a(3) = 8 because 8 + 8 = 16 = 4^2.
MATHEMATICA
a=1; lst={0, a}; Do[a=n^2-a; AppendTo[lst, a], {n, 3, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
Table[(n+2)*(n+1)/2 + 2*(-1)^n, {n, 0, 25}] (* G. C. Greubel, Oct 12 2016 *)
PROG
(Magma) [0] cat [(n+2)*(n+1)/2+2*(-1)^n: n in [1..60]]; // Vincenzo Librandi, Feb 14 2015
(PARI) concat(0, Vec(x*(1+6*x-8*x^2+3*x^3)/((1-x)^3*(1+x)) + O(x^60))) \\ Colin Barker, Oct 13 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alexander R. Povolotsky, Dec 11 2007, Apr 02 2008
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Dec 17 2008
STATUS
approved