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a(n) = numerator((n^2 + 3*n + 2)/n^3).
3

%I #32 Feb 26 2017 03:12:16

%S 6,3,20,15,42,7,72,45,110,33,156,91,210,30,272,153,342,95,420,231,506,

%T 69,600,325,702,189,812,435,930,124,1056,561,1190,315,1332,703,1482,

%U 195,1640,861,1806,473,1980,1035,2162,282,2352,1225,2550,663,2756,1431,2970,385

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

%H Colin Barker, <a href="/A276805/b276805.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).

%F From _Colin Barker_, Sep 18 2016: (Start)

%F a(n) = 3*a(n-8)-3*a(n-16)+a(n-24) for n>24.

%F G.f.: x*(6 +3*x +20*x^2 +15*x^3 +42*x^4 +7*x^5 +72*x^6 +45*x^7 +92*x^8 +24*x^9 +96*x^10 +46*x^11 +84*x^12 +9*x^13 +56*x^14 +18*x^15 +30*x^16 +5*x^17 +12*x^18 +3*x^19 +2*x^20 +x^23) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3).

%F (End)

%F a(n) = a(n-8)*(n^2+3*n+2)/(n^2-13*n+42), for n>8. - _Gionata Neri_, Feb 25 2017

%e a(1) = numerator((n^2 + 3*n + 2)/n^3) = 1^2+3*1+2/1^3 = 6.

%t Table[Numerator[(n^2 + 3*n + 2)/n^3], {n, 1, 100}]

%o (PARI) a(n) = numerator((n^2 + 3*n + 2)/n^3); \\ _Michel Marcus_, Sep 18 2016

%o (PARI) Vec(x*(6 +3*x +20*x^2 +15*x^3 +42*x^4 +7*x^5 +72*x^6 +45*x^7 +92*x^8 +24*x^9 +96*x^10 +46*x^11 +84*x^12 +9*x^13 +56*x^14 +18*x^15 +30*x^16 +5*x^17 +12*x^18 +3*x^19 +2*x^20 +x^23) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3) + O(x^100)) \\ _Colin Barker_, Oct 20 2016

%Y Cf. A277542.

%K nonn,easy,frac

%O 1,1

%A _Pietro Tiaraju Giavarina dos Santos_, Sep 17 2016