A046163 -id:A046163 - cp's OEIS frontend

A153171 First differences of A046163.

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

Offset: 0

Paul Curtz, Dec 20 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,2,2,2,-1,-1,-1).

Cf. A046163.

Differences[Table[Denominator[(n-1)^2/(n^2+n+1)],{n,1,50}]] (* Vaclav Kotesovec, Nov 04 2014 *)

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

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

Edited by N. J. A. Sloane, Dec 20 2008

More terms from Colin Barker, Nov 04 2014

A158622 Numerator of the reduced fraction A158620(n)/A158621(n).

Original entry on oeis.org

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

Offset: 2

Jonathan Vos Post, Mar 23 2009

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, ...
Is this the same as A046163? - R. J. Mathar, Mar 27 2009
Apparently a(n) = A130770(n) for 2 <= n <= 53. - Georg Fischer, Oct 24 2018

			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.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A001093, A016921, A068601, A130770, A158620-A158621, A158623.

A158622 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; numer(%) ; end: seq(A158622(n),n=2..100) ; # R. J. Mathar, Mar 27 2009

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 *)

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).
A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). - R. J. Mathar, Mar 27 2009
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

More terms from R. J. Mathar, Mar 27 2009

A046162 Reduced numerators of (n-1)^2/(n^2 + n + 1).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

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

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Routh's Theorem.

Cf. A046163 (denominators), A147560.

[Numerator((n-1)^2/3): n in [1..70]]; // G. C. Greubel, Oct 27 2022

seq(numer((n-1)^2/(n^2+n+1)), n=1..51) ; # Zerinvary Lajos, Jun 04 2008
seq(denom(3/n^2-2), n=0..76) ; # Zerinvary Lajos, Jun 04 2008

a[n_] := Numerator[(n - 1)^2/(n^2 + n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 11 2022 *)

[numerator((n-1)^2/3) for n in range(1,71)] # G. C. Greubel, Oct 27 2022

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) = (n-1)^2/3 if n-1 == 0 (mod 3), (n-1)^2 otherwise. - David W. Wilson, Jun 12 2005, corrected by Robert Israel, Apr 28 2017
From Amiram Eldar, Aug 11 2022: (Start)
a(n) = numerator((n-1)^2/3).
Sum_{n>=2} 1/a(n) = 11*Pi^2/54. (End)
From Amiram Eldar, Dec 30 2022: (Start)
With offset 0, Dirichlet g.f.: zeta(s-2)*(1-6/3^s).
Sum_{k=1..n} a(k) ~ 7*n^3/27. (End)

A130770 One third of the least common multiple of 3 and n^2+n+1.

Original entry on oeis.org

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

Offset: 0

W. Neville Holmes, Jul 14 2007

This is a subset of A051176 and is also one third of A130723.

			a(4)=7 because 4^2+4+1 =21, the LCM of 3 and 21 is 21 and 21/3=7.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -3, 0, 0, 1).

Cf. A051176, A130723.

[Lcm(3,n^2+n+1)/3: n in [0..50]]; // G. C. Greubel, Oct 26 2017

seq(denom((n-1)^2/(n^2+n+1)), n=0..52) ; # Zerinvary Lajos, Jun 04 2008

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 *)

for(n=0,50, print1(lcm(3, n^2 + n +1)/3, ", ")) \\ G. C. Greubel, Oct 26 2017

Conjecture: a(n) = A046163(n), n>0. - R. J. Mathar, Jun 13 2008

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

A153186 Period 9: repeat 1,7,4,7,4,7,1,1,1.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 20 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A046163.

PadRight[{},120,{1,7,4,7,4,7,1,1,1}] (* Harvey P. Dale, Sep 14 2019 *)

a(n)=[1,7,4,7,4,7,1,1,1][n%9+1] \\ Charles R Greathouse IV, Jul 13 2016

Edited by N. J. A. Sloane, Dec 25 2008

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A153171 First differences of A046163.

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A158622 Numerator of the reduced fraction A158620(n)/A158621(n).

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A046162 Reduced numerators of (n-1)^2/(n^2 + n + 1).

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A130770 One third of the least common multiple of 3 and n^2+n+1.

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A153186 Period 9: repeat 1,7,4,7,4,7,1,1,1.

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