%I #49 Sep 08 2022 08:45:05
%S 24,120,336,720,1320,2184,3360,4896,6840,9240,12144,15600,19656,24360,
%T 29760,35904,42840,50616,59280,68880,79464,91080,103776,117600,132600,
%U 148824,166320,185136,205320,226920,249984,274560,300696,328440,357840
%N a(n) = (2*n+2)*(2*n+3)*(2*n+4) = 24*A000330(n+1).
%C sqrt((Sum_{k=0..n} 2*a(k)) + 1) = A056220(n+2). - _Doug Bell_, Mar 09 2009
%C Second leg of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + a(n-1)^2 = A007204(n)^2. - _Martin Renner_, Nov 12 2011
%C 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
%D Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53.
%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 190.
%D Jolley, Summation of Series, Dover (1961).
%D Konrad Knopp, Theory and application of infinite series, Dover, p. 269
%H Vincenzo Librandi, <a href="/A069074/b069074.txt">Table of n, a(n) for n = 0..10000</a>
%H Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F Sum_{n>=0} (-1)^n/a(n) = (Pi-3)/4 = 0.03539816339... [Jolley, eq. 244]
%F Sum_{n>=0} 1/a(n) = 3/4 - log(2) = 0.05685281... [Jolley, eq. 249]
%F G.f.: ( 24+24*x ) / (x-1)^4. - _R. J. Mathar_, Oct 03 2011
%t LinearRecurrence[{4,-6,4,-1},{24,120,336,720},40] (* _Harvey P. Dale_, Apr 10 2017 *)
%o (Magma) [(2*n+2)*(2*n+3)*(2*n+4): n in [0..40]]; // _Vincenzo Librandi_, Oct 04 2011
%o (PARI) a(n)=6*binomial(2*n+4,3) \\ _Charles R Greathouse IV_, Mar 21 2015
%Y Cf. A001844. A001844(n+1)^2 - a(n) and A001844(n+1)^2 + a(n) are both square numbers. - _Doug Bell_, Mar 08 2009
%Y 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
%K easy,nonn
%O 0,1
%A _Benoit Cloitre_, Apr 05 2002