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6, 6, -6, 24, 12, -24, 54, 18, -54, 96, 24, -96, 150, 30, -150, 216, 36, -216, 294, 42, -294, 384, 48, -384, 486, 54, -486, 600, 60, -600, 726, 66, -726, 864, 72, -864, 1014, 78, -1014, 1176, 84, -1176, 1350, 90, -1350, 1536, 96, -1536, 1734, 102, -1734
FORMULA
G.f.: 6*(x^4+2*x^3+x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^3). - Colin Barker, Nov 04 2014
MATHEMATICA
Differences[Table[Denominator[(n-1)^2/(n^2+n+1)], {n, 1, 50}]] (* Vaclav Kotesovec, Nov 04 2014 *)
PROG
(PARI) Vec(6*(x^4+2*x^3+x^2+2*x+1)/((x-1)^2*(x^2+x+1)^3) + O(x^100)) \\ Colin Barker, Nov 04 2014
7, 13, 7, 31, 43, 19, 73, 91, 37, 133, 157, 61, 211, 241, 91, 307, 343, 127, 421, 463, 169, 553, 601, 217, 703, 757, 271, 871, 931, 331, 1057, 1123, 397, 1261, 1333, 469, 1483, 1561, 547, 1723, 1807, 631, 1981, 2071, 721, 2257, 2353, 817, 2551, 2653, 919, 2863
COMMENTS
A158620(n) = Product_{k=2..n} (k^3-1). A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/ A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/ A158621(n). The reduced fractions are 7/9, 13/18, 7/10, 31/45, 43/63, 19/28, 73/108, 91/135, 37/55, 133/198, ...
FORMULA
Numerator of (Product_{k=2..n} (k^3-1))/Product_{k=2..n} (k^3+1) = numerator of Product_{k=2..n} A068601(k)/ A001093(k). Empirical g.f.: -x^2*(x^8 + x^7 + x^6 - 2*x^5 + 4*x^4 + 10*x^3 + 7*x^2 + 13*x + 7) / ((x-1)^3*(x^2 + x + 1)^3). - Colin Barker, May 09 2013
EXAMPLE
a(2) = 7 = numerator of (2^3-1)/2^3+1 = 7/9.
a(3) = 13 = numerator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18.
a(4) = 7 = = numerator of ((2^3-1)*(3^3-1)*(4^3-1))/((2^3+1)*(3^3+1)*(4^3+1)) = (7 * 26 * 63)/(9 * 28 * 65) = 11466/16380 = 7/10.
a(5) = 31 = numerator of ((2^3-1)(3^3-1)(4^3-1)(5^3-1))/((2^3+1)(3^3+1)(4^3+1)(5^3+1)) = 1421784/2063880 = 31/45.
MATHEMATICA
Table[Product[k^3-1, {k, 2, n}]/Product[k^3+1, {k, 2, n}], {n, 2, 60}]//Numerator (* Harvey P. Dale, Feb 26 2020 *)
Reduced numerators of (n-1)^2/(n^2 + n + 1).
+10
3
0, 1, 4, 3, 16, 25, 12, 49, 64, 27, 100, 121, 48, 169, 196, 75, 256, 289, 108, 361, 400, 147, 484, 529, 192, 625, 676, 243, 784, 841, 300, 961, 1024, 363, 1156, 1225, 432, 1369, 1444, 507, 1600, 1681, 588, 1849, 1936, 675, 2116, 2209, 768, 2401, 2500
COMMENTS
Arises in Routh's theorem.
With offset 0, multiplicative with a(3^e) = 3^(2e-1), a(p^e) = p^(2e) otherwise. - David W. Wilson, Jun 12 2005, corrected by Robert Israel, Apr 28 2017
FORMULA
G.f.: x^2*(1 + 4*x + 3*x^2 + 13*x^3 + 13*x^4 + 3*x^5 + 4*x^6 + x^7)/(1 - x^3)^3.
a(n) = numerator((n-1)^2/3).
Sum_{n>=2} 1/a(n) = 11*Pi^2/54. (End)
With offset 0, Dirichlet g.f.: zeta(s-2)*(1-6/3^s).
Sum_{k=1..n} a(k) ~ 7*n^3/27. (End)
MATHEMATICA
a[n_] := Numerator[(n - 1)^2/(n^2 + n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 11 2022 *)
PROG
(Magma) [Numerator((n-1)^2/3): n in [1..70]]; // G. C. Greubel, Oct 27 2022 (SageMath) [numerator((n-1)^2/3) for n in range(1, 71)] # G. C. Greubel, Oct 27 2022
One third of the least common multiple of 3 and n^2+n+1.
+10
2
1, 1, 7, 13, 7, 31, 43, 19, 73, 91, 37, 133, 157, 61, 211, 241, 91, 307, 343, 127, 421, 463, 169, 553, 601, 217, 703, 757, 271, 871, 931, 331, 1057, 1123, 397, 1261, 1333, 469, 1483, 1561, 547, 1723, 1807, 631, 1981, 2071, 721, 2257, 2353, 817, 2551, 2653, 919
FORMULA
a(n) = 3*a(n-3)-3*a(n-6)+a(n-9), with a(0)=1, a(1)=1, a(2)=7, a(3)=13, a(4)=7, a(5)=31, a(6)=43, a(7)=19, a(8)=73. - Harvey P. Dale, Apr 10 2014
EXAMPLE
a(4)=7 because 4^2+4+1 =21, the LCM of 3 and 21 is 21 and 21/3=7.
MATHEMATICA
Table[LCM[3, n^2+n+1]/3, {n, 0, 60}] (* or *) LinearRecurrence[ {0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 7, 13, 7, 31, 43, 19, 73}, 60] (* Harvey P. Dale, Apr 10 2014 *)
PROG
(PARI) for(n=0, 50, print1(lcm(3, n^2 + n +1)/3, ", ")) \\ G. C. Greubel, Oct 26 2017 (Magma) [Lcm(3, n^2+n+1)/3: n in [0..50]]; // G. C. Greubel, Oct 26 2017
Period 9: repeat 1,7,4,7,4,7,1,1,1.
+10
0
1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 1, 1
MATHEMATICA
PadRight[{}, 120, {1, 7, 4, 7, 4, 7, 1, 1, 1}] (* Harvey P. Dale, Sep 14 2019 *)
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