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A256833 a(n) = (4*n+3)*(4*n+2). 0
6, 42, 110, 210, 342, 506, 702, 930, 1190, 1482, 1806, 2162, 2550, 2970, 3422, 3906, 4422, 4970, 5550, 6162, 6806, 7482, 8190, 8930, 9702, 10506, 11342, 12210, 13110, 14042, 15006, 16002, 17030, 18090, 19182, 20306, 21462, 22650, 23870, 25122, 26406 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Since 0 = Sin(Pi) = Sum_{n>=0}(-1)^n*Pi^(2n+1)/(2n+1)!, we can move the negative terms to the other side of the equation to get: Sum_{n>=0} Pi^(4n+1)/(4n+1)! = Sum_{n>=0}Pi^(4n+3)/(4n+3)!.

Now, if we let f(n) = Pi^(4n+1)/(4n+1)!, then the previous equation can be written as Sum_{n>=0}f(n) = Sum_{n>=0}(Pi^2/((4*n+3)*(4*n+2)))*f(n); a(n) is the n-th denominator on the right hand side.

LINKS

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 16*n^2 + 20*n + 6.

a(n) = 2*A033567(n+1).

G.f.: (6+24*x+2*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 12 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Vincenzo Librandi, Apr 12 2015

a(n) = A016825(n)*A004767(n). - Tom Edgar, Apr 12 2015

a(n) = A002378(4*n+2) = 2*A000217(4*n+2). - Ivan N. Ianakiev, Apr 17 2015

MATHEMATICA

CoefficientList[Series[(6 + 24 x + 2 x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 12 2015 *)

PROG

(MAGMA) [16*n^2 + 20*n + 6: n in [0..40]]; // Vincenzo Librandi, Apr 12 2015

(PARI) vector(50, n, (4*n-1)*(4*n-2)) \\ Derek Orr, Apr 13 2015

CROSSREFS

Sequence in context: A044489 A211616 A153786 * A164016 A147811 A046763

Adjacent sequences:  A256830 A256831 A256832 * A256834 A256835 A256836

KEYWORD

nonn,easy

AUTHOR

Bruce Zimov, Apr 10 2015

EXTENSIONS

More terms from Vincenzo Librandi, Apr 12 2015

STATUS

approved

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Last modified January 20 10:23 EST 2018. Contains 297960 sequences.