A162668 - OEIS

%I #39 Nov 03 2022 05:43:29

%S 0,8,40,120,280,560,1008,1680,2640,3960,5720,8008,10920,14560,19040,

%T 24480,31008,38760,47880,58520,70840,85008,101200,119600,140400,

%U 163800,190008,219240,251720,287680,327360,371008,418880,471240,528360,590520

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

%C 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 Vincenzo Librandi, <a href="/A162668/b162668.txt">Table of n, a(n) for n = 0..10000</a>

%H Diego Marques, <a href="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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F From _R. J. Mathar_, Jul 13 2009: (Start)

%F a(n) = 8*A000332(n+3).

%F G.f.: 8*x/(1-x)^5. (End)

%F 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 E.g.f.: x*(24 + 36*x + 12*x^2 + x^3)*exp(x)/3. - _G. C. Greubel_, Aug 27 2019

%F From _Amiram Eldar_, Nov 03 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 1/6.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2) - 8/3.

%F Product_{n>=1} 1-1/a(n) = 4*cos(sqrt(13)*Pi/2)*cosh(sqrt(3)*Pi/2)/(3*Pi^2). (End)

%e G.f. = 8*x + 40*x^2 + 120*x^3 + 280*x^4 + 560*x^5 + ... - _Michael Somos_, Oct 19 2022

%p seq(8*binomial(n+3, 4), n=0..40); # _G. C. Greubel_, Aug 27 2019

%t 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 8*Binomial[Range[40]+2, 4] (* _G. C. Greubel_, Aug 27 2019 *)

%o (Magma) [n*(n+1)*(n+2)*(n+3)/3: n in [0..40] ];

%o (PARI) binomial(n+3,4)/8 \\ _Charles R Greathouse IV_, Jan 11 2012

%o (Sage) [8*binomial(n+3, 4) for n in (0..40)] # _G. C. Greubel_, Aug 27 2019

%o (GAP) List([0..40], n-> 8*Binomial(n+3,4)); # _G. C. Greubel_, Aug 27 2019

%Y Cf. A000332, A162669.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Jul 10 2009

%E Definition factorized, offset corrected by _R. J. Mathar_, Jul 13 2009