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Revision History for A322171

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
newer changes | Showing entries 11-20 | older changes

A322171

Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).
(history; published version)

#19 by Muniru A Asiru at Thu Dec 06 10:23:27 EST 2018

MAPLE

seq(coeff(series(x*(x^3+x^2+5*x+3)/((1-x)^2*(1+x^2)), x, n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 06 2018

STATUS

proposed

editing

#18 by Vincenzo Librandi at Thu Dec 06 04:39:39 EST 2018

STATUS

editing

proposed

#17 by Vincenzo Librandi at Thu Dec 06 04:39:19 EST 2018

FORMULA

G.f.: x*(x^3 + x^2 + 5*x + 3) / ((x - 1)^2 *(x^2 + 1)). - Vincenzo Librandi, Dec 06 2018

MATHEMATICA

CoefficientList[Series[(x^3 + x^2 + 5x 5 x + 3)/((x - 1)^2 (x^2 + 1)), {x, 0, 50}], x] (* or *)

a[n_]:= (1/2)* (10* n - (1 + 2 * I)* (-I)^n - (1 - 2* I)* I^n); Simplify[Array[a, 50]] (* Stefano Spezia, Nov 29 2018 *)

LinearRecurrence[{2, -2, 2, -1}, {3, 11, 17, 19}, 60] (* Vincenzo Librandi, Dec 06 2018 *)

PROG

(MAGMA) I:=[3, 11, 17, 19]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Dec 06 2018

STATUS

proposed

editing

#16 by Andrew Howroyd at Thu Nov 29 22:15:02 EST 2018

STATUS

editing

proposed

Discussion

Fri Nov 30

00:20

Stefano Spezia: You are welcome Andrew

09:14

Mark A. Thomas: Andrew, no particular math reason for the g. f.. I was interested in the very shallow sine wave like structure this presented. I am not sure if there are very many of these in the OEIS, perhaps I am wrong.

#15 by Andrew Howroyd at Thu Nov 29 22:13:04 EST 2018

NAME

Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

FORMULA

a(n) = 5*n - A228826(n-1). - Andrew Howroyd, Nov 29 2018

CROSSREFS

Cf. A228826.

Discussion

Thu Nov 29

22:14

Andrew Howroyd: Expansion needed a multiplier of x for this offset.

#14 by Andrew Howroyd at Thu Nov 29 22:03:53 EST 2018

LINKS

<a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, -2, 2, -1)

KEYWORD

nonn,easy,changed

STATUS

proposed

editing

Discussion

Thu Nov 29

22:06

Andrew Howroyd: Mark is there a reason why this particular g.f is of interest? Additional information or cross-references or references would be useful.

#13 by Andrew Howroyd at Thu Nov 29 22:00:44 EST 2018

STATUS

editing

proposed

#12 by Andrew Howroyd at Thu Nov 29 21:57:33 EST 2018

NAME

Expansion of (x^3 + 5*x + x^2 + 5x +x^3)/((x-1 - x)^2*(1 + x^2+1)).

FORMULA

a(n) = (1/2)*(10*n - (1+2*i)*(-i)^n - (1-2*i)*i^n), where i is the imaginary unity. - Corrected by _Stefano Spezia_, Nov 29 2018

a(n) = 5*n - 2*sin(Pi*n/2) - cos(Pi*n/2).

PROG

(PARI) Vec((3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)) + O(x^60)) \\ Andrew Howroyd, Nov 29 2018

STATUS

proposed

editing

Discussion

Thu Nov 29

22:00

Andrew Howroyd: Thanks Stefano for finding and fixing the the formula. Since this isn't yet published, its probably too soon to be recording the correction. The sin/cos stuff was a second formula which I have restored.

#11 by Stefano Spezia at Thu Nov 29 15:10:51 EST 2018

STATUS

editing

proposed

#10 by Stefano Spezia at Thu Nov 29 15:10:48 EST 2018

FORMULA

5*x-2*sin(Pi*x/2)-cos(Pi*x/2).

STATUS

proposed

editing