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A100886 Expansion of x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2)). 3
0, 1, 3, 3, 5, 10, 14, 23, 39, 61, 99, 162, 260, 421, 683, 1103, 1785, 2890, 4674, 7563, 12239, 19801, 32039, 51842, 83880, 135721, 219603, 355323, 574925, 930250, 1505174, 2435423, 3940599, 6376021, 10316619, 16692642, 27009260, 43701901 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence was investigated in cooperation with Paul Barry. Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("tes"). (1/2)(a(n) + A100887(n) - A100888(n)) gives A061347(n+3).

LINKS

Table of n, a(n) for n=0..37.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

FORMULA

a(n) = (L(n+1)-A061347(n))/2, L=A000032; a(n)=a(n-2)+2a(n-3)+a(n-4), a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 3.

a(n) = n*sum(j=1,floor(n/2), binomial(2*j,n-2*j)/(2*j) ). - Vladimir Kruchinin, Apr 09 2011

MATHEMATICA

a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 3; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 36}]

(* Or *) CoefficientList[ Series[x(1 + 3x + 2x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 36}], x] (* Robert G. Wilson v, Nov 26 2004 *)

LinearRecurrence[{0, 1, 2, 1}, {0, 1, 3, 3}, 40] (* Harvey P. Dale, Apr 04 2016 *)

PROG

(Maxima) a(n):=n*sum(binomial(k, n-k)*(if oddp(k) then 0 else 1/k), k, 1, n) /* Vladimir Kruchinin, Apr 09 2011 */

(PARI)

A100886(n)=n*sum(j=1, n\2, k=2*j; binomial(k, n-k)/k);

vector(66, n, A100886(n)) /* show terms */ /* Joerg Arndt, Apr 09 2011 */

(PARI)

Vec(x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2))+O(x^66)) /* show terms starting with 1 */ /* Joerg Arndt, Apr 09 2011 */

(MAGMA) I:=[0, 1, 3, 3]; [n le 4 select I[n] else Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 30 2015

CROSSREFS

Cf. A087204, A100887, A100888, A100889, A100890.

Sequence in context: A174102 A217521 A252943 * A072337 A132751 A218354

Adjacent sequences:  A100883 A100884 A100885 * A100887 A100888 A100889

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Nov 21 2004

EXTENSIONS

More terms from Robert G. Wilson v, Nov 26 2004

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.