A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

0, 1, 2, 7, 18, 49, 128, 337, 882, 2311, 6050, 15841, 41472, 108577, 284258, 744199, 1948338, 5100817, 13354112, 34961521, 91530450, 239629831, 627359042, 1642447297, 4299982848, 11257501249, 29472520898, 77160061447, 202007663442, 528862928881, 1384581123200

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

Cf. A192872, A192232, A192744.

Sequence in context: A072338 A182197 A022726 * A017925 A030236 A074141

Adjacent sequences: A192870 A192871 A192872 * A192874 A192875 A192876

Keyword

nonn,easy

Author

Clark Kimberling, Jul 11 2011

Extensions

More terms from Colin Barker, Apr 01 2014

Status

approved