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A069072 a(n) = (2n+1)*(2n+2)*(2n+3). 7
6, 60, 210, 504, 990, 1716, 2730, 4080, 5814, 7980, 10626, 13800, 17550, 21924, 26970, 32736, 39270, 46620, 54834, 63960, 74046, 85140, 97290, 110544, 124950, 140556, 157410, 175560, 195054, 215940, 238266, 262080, 287430, 314364, 342930 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Terms are areas of primitive Pythagorean triangles whose odd sides differ by 2; e.g., the triangle with sides 8,15,17 has area 60. - Lekraj Beedassy, Apr 18 2003

Using (n, n+1), (n, n+2), and (n+1, n+2) to generate three unreduced Pythagorean triangles gives a sum of the areas for all three to be (2*n+1)*(2*n+2)*(2*n+3), which are three consecutive numbers. - J. M. Bergot, Aug 22 2011

REFERENCES

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 190.

Jolley, Summation of Series, Oxford (1961).

Konrad Knopp, Theory and application of infinite series, Dover, p. 269.

LINKS

Table of n, a(n) for n=0..34.

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")

S. Ramanujan, Notebook entry

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

log(2) - 1/2 = sum_{n>=0}, 1/a(n); (1/2)*(1-log(2)) = sum_{n>=0} (-1)^n/a(n). [Jolley eq 236 and 237]

sum_{n>=0} x^n/a(n) = ((1+x)/sqrt(x)*log((1+sqrt x)/(1-sqrt x)) + 2*log(1-x)-2)/(4x). [Jolley eq 280 for 0<x<1]

sum_{n>=0} (-x)^n/a(n) = (1-log(1+x) -(1-x)/sqrt(x)*arctan(x))/(2x). [Jolley eq 281 for 0<x<=1]

a(n) = 6*A000447(n+1). - Lekraj Beedassy, Apr 18 2003

G.f.: 6*(1 + 6*x + x^2) / (x-1)^4 . - R. J. Mathar, Jun 09 2013

a(0)=6, a(1)=60, a(2)=210, a(3)=504, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 08 2013

a(n) = 2*A035328(n+1). - J. M. Bergot, Jan 02 2015

MATHEMATICA

Array[Times@@(2#+{1, 2, 3})&, 40, 0] (* or *) LinearRecurrence[{4, -6, 4, -1}, {6, 60, 210, 504}, 40] (* Harvey P. Dale, Dec 08 2013 *)

PROG

(PARI) a(n)=(2*n+1)*(2*n+2)*(2*n+3) \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A097321, A069140.

Sequence in context: A126576 A232181 A121287 * A256442 A296317 A292061

Adjacent sequences:  A069069 A069070 A069071 * A069073 A069074 A069075

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Apr 05 2002

STATUS

approved

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Last modified November 12 16:45 EST 2019. Contains 329058 sequences. (Running on oeis4.)