OFFSET
0,3
COMMENTS
The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + (x^2)*p(n-1,x) + 1. See A192872.
First differences give A236428. - Richard R. Forberg, Feb 23 2014
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).
FORMULA
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
G.f.: x*(x^2-x+1) / ((1-x)*(1+x)*(x^2-3*x+1)). - Colin Barker, Apr 01 2014
a(n) = (1/10) * (4L(2*n) - 3*(-1)^n - 5), with L(n) the Lucas numbers (A000032). - Ralf Stephan, Apr 06 2014
a(-n) = a(n) for all n in Z. - Michael Somos, Apr 08 2014
EXAMPLE
The coefficients of all the polynomials p(n,x) are Fibonacci numbers (A000045). The first 6 and their reductions:
p(0,x) = 1 -> 1
p(1,x) = x -> x
p(2,x) = 1 +2*x^2 -> 3 +2*x
p(3,x) = 1 +x +3*x^3 -> 4 +7*x
p(4,x) = 1 +x +2*x^2 +5*x^4 -> 13 +18*x
p(5,x) = 1 +x +2*x^2 +3*x^3 +8*x^5 -> 30 +49*x
G.f. = x + 2*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 128*x^6 + 337*x^7 + ...
MAPLE
seq(coeff(series(x*(x^2-x+1)/((1-x)*(1+x)*(x^2-3*x+1)), x, n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Jan 08 2019
MATHEMATICA
(See A192872.)
a[ n_] := SeriesCoefficient[ x * (1 - x + x^2) / ((1 - x^2) * (1 - 3*x + x^2)), {x, 0, Abs @ n}]; (* Michael Somos, Apr 08 2014 *)
LinearRecurrence[{3, 0, -3, 1}, {0, 1, 2, 7}, 40] (* G. C. Greubel, Jan 07 2019 *)
PROG
(PARI) concat(0, Vec(-x*(x^2-x+1)/((x-1)*(x+1)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Apr 01 2014
(Magma) I:=[0, 1, 2, 7]; [n le 4 select I[n] else 3*Self(n-1) - 3*Self(n-3) +Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 07 2019
(Sage) (x*(x^2-x+1)/((1-x^2)*(x^2-3*x+1))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 07 2019
(GAP) a:=[0, 1, 2, 7];; for n in [5..40] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 07 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 11 2011
EXTENSIONS
More terms from Colin Barker, Apr 01 2014
STATUS
approved