A069074 a(n) = (2*n+2)*(2*n+3)*(2*n+4) = 24*A000330(n+1).

24, 120, 336, 720, 1320, 2184, 3360, 4896, 6840, 9240, 12144, 15600, 19656, 24360, 29760, 35904, 42840, 50616, 59280, 68880, 79464, 91080, 103776, 117600, 132600, 148824, 166320, 185136, 205320, 226920, 249984, 274560, 300696, 328440, 357840

Offset

0,1

Comments

sqrt((Sum_{k=0..n} 2*a(k)) + 1) = A056220(n+2). - Doug Bell, Mar 09 2009

Second leg of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + a(n-1)^2 = A007204(n)^2. - Martin Renner, Nov 12 2011

Numbers which are both the sum of 2*n + 4 consecutive odd integers and the sum of the 2*n + 2 immediately higher consecutive odd integers. In general, let f(k,n) = 3*k^3*A000330(n). Then f(k,n) is both the sum of k*n + k consecutive terms from the arithmetic progression with first term A000217(k) and constant difference k and the immediately higher k*n terms from the same progression. When k = 1, f(k,n) = A059270(n). - Charlie Marion, Aug 23 2021

References

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53.

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

Jolley, Summation of Series, Dover (1961).

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

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

Formula

Sum_{n>=0} (-1)^n/a(n) = (Pi-3)/4 = 0.03539816339... [Jolley, eq. 244]

Sum_{n>=0} 1/a(n) = 3/4 - log(2) = 0.05685281... [Jolley, eq. 249]

G.f.: ( 24+24*x ) / (x-1)^4. - R. J. Mathar, Oct 03 2011

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {24, 120, 336, 720}, 40] (* Harvey P. Dale, Apr 10 2017 *)

Prog

(Magma) [(2*n+2)*(2*n+3)*(2*n+4): n in [0..40]]; // Vincenzo Librandi, Oct 04 2011
(PARI) a(n)=6*binomial(2*n+4, 3) \\ Charles R Greathouse IV, Mar 21 2015

Crossrefs

Cf. A001844. A001844(n+1)^2 - a(n) and A001844(n+1)^2 + a(n) are both square numbers. - Doug Bell, Mar 08 2009

Cf. A000466. a(n) = Sum_{k=0..2n+3} (A000466(n+1) + 2k) which is the sum of 2n+4 consecutive odd integers starting at A000466(n+1). - Doug Bell, Mar 08 2009

Sequence in context: A256629 A114200 A229567 * A360389 A059775 A372497

Adjacent sequences: A069071 A069072 A069073 * A069075 A069076 A069077

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 05 2002

Status

approved