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A162668 a(n) = n*(n+1)*(n+2)*(n+3)/3.

%I A162668 #40 Jan 05 2025 19:51:39
%S A162668 0,8,40,120,280,560,1008,1680,2640,3960,5720,8008,10920,14560,19040,
%T A162668 24480,31008,38760,47880,58520,70840,85008,101200,119600,140400,
%U A162668 163800,190008,219240,251720,287680,327360,371008,418880,471240,528360,590520
%N A162668 a(n) = n*(n+1)*(n+2)*(n+3)/3.
%C A162668 a(n+3) is the number of equivalence classes of n-tuples from the set {1,0,-1} where the number of nonzero elements is 4 and two n-tuples are equivalent if they are negatives of each other. - _Michael Somos_, Oct 19 2022
%H A162668 Vincenzo Librandi, <a href="/A162668/b162668.txt">Table of n, a(n) for n = 0..10000</a>
%H A162668 Diego Marques, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/51-1/MarquesOrderConsecLucas.pdf">The order of appearance of the product of consecutive Lucas numbers</a>, Fibonacci Quarterly, 51 (2013), 38-43. - From _N. J. A. Sloane_, Mar 06 2013
%H A162668 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A162668 From _R. J. Mathar_, Jul 13 2009: (Start)
%F A162668 a(n) = 8*A000332(n+3).
%F A162668 G.f.: 8*x/(1-x)^5. (End)
%F A162668 For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} sin(x)^7 * cos(x)^(2*n-1) dx). - _Francesco Daddi_, Aug 02 2011
%F A162668 E.g.f.: x*(24 + 36*x + 12*x^2 + x^3)*exp(x)/3. - _G. C. Greubel_, Aug 27 2019
%F A162668 From _Amiram Eldar_, Nov 03 2022: (Start)
%F A162668 Sum_{n>=1} 1/a(n) = 1/6.
%F A162668 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2) - 8/3.
%F A162668 Product_{n>=1} 1-1/a(n) = 4*cos(sqrt(13)*Pi/2)*cosh(sqrt(3)*Pi/2)/(3*Pi^2). (End)
%e A162668 G.f. = 8*x + 40*x^2 + 120*x^3 + 280*x^4 + 560*x^5 + ... - _Michael Somos_, Oct 19 2022
%p A162668 seq(8*binomial(n+3, 4), n=0..40); # _G. C. Greubel_, Aug 27 2019
%t A162668 Table[n*(n+1)*(n+2)*(n+3)/3, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009, modified by _G. C. Greubel_, Aug 27 2019 *)
%t A162668 8*Binomial[Range[40]+2, 4] (* _G. C. Greubel_, Aug 27 2019 *)
%o A162668 (Magma) [n*(n+1)*(n+2)*(n+3)/3: n in [0..40] ];
%o A162668 (PARI) binomial(n+3,4)/8 \\ _Charles R Greathouse IV_, Jan 11 2012
%o A162668 (Sage) [8*binomial(n+3, 4) for n in (0..40)] # _G. C. Greubel_, Aug 27 2019
%o A162668 (GAP) List([0..40], n-> 8*Binomial(n+3,4)); # _G. C. Greubel_, Aug 27 2019
%Y A162668 Cf. A000332, A162669.
%K A162668 nonn,easy
%O A162668 0,2
%A A162668 _Vincenzo Librandi_, Jul 10 2009
%E A162668 Definition factorized, offset corrected by _R. J. Mathar_, Jul 13 2009