OFFSET
1,1
COMMENTS
Polynomials f(x) have the following property: f(x + n*f(x)) is congruent to f(x); here n is an integer.
This can be proved by Taylor's theorem.
After rationalization of the denominator, the quotient q(n,x) = f(x + n*f(x))/f(x) consists of two parts:
a) a rational integer and b) an irrational part.
The present sequence is the integer part for f(x) = x^2 + 3x + 13 and x = sqrt(2), i.e., q(n,x) = a(n) + sqrt(2)*A045944(n).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(19 + 10*x + x^2)/(1-x)^3. - R. J. Mathar, Sep 29 2009
From G. C. Greubel, Apr 08 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (15*x^2 + 18*x + 1)*exp(x). (End)
EXAMPLE
When we substitute sqrt(2) for x in the quadratic expression x^2 + 3x + 13 we get 15 + 3*sqrt(2).
sqrt(2) + (15 + 3*sqrt(2)) = (15 + 4*sqrt(2)). When this is substituted in f(x) we get 270 + 132*sqrt(2).
(270 + 132*sqrt(2))/(15+3*sqrt(2)) = 19 + 5*sqrt(2).
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {19, 67, 145}, 100] (* G. C. Greubel, Apr 08 2016 *)
Table[15n^2+3n+1, {n, 50}] (* Harvey P. Dale, Mar 14 2020 *)
PROG
(PARI) a(n)=15*n^2+3*n+1 \\ Charles R Greathouse IV, Sep 28 2011
(Magma) [15*n^2 + 3*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 29 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
A.K. Devaraj, Sep 28 2009
EXTENSIONS
Definition simplified, sequence extended by R. J. Mathar, Sep 29 2009
STATUS
approved