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A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 20 02:46 EDT 2019. Contains 322291 sequences. (Running on oeis4.)