A306367 -id:A306367 - cp's OEIS frontend

A129202 Denominator of 3*(3+(-1)^n) / (n+1)^2.

1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54, 361, 200, 147, 242, 529, 96, 625, 338, 243, 392, 841, 150, 961, 512, 363, 578, 1225, 216, 1369, 722, 507, 800, 1681, 294, 1849, 968, 675, 1058, 2209, 384, 2401, 1250, 867, 1352, 2809, 486

Offset: 0

Paul Barry, Apr 03 2007

A divisibility sequence, that is, if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, A note on the sequence of numerators of a rational function , Feb 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Cf. A026741, A051176, A129196, A129197 (numerators), A060789.

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

A129202:=n->numer((n+1)/2)*numer((n+1)/3): seq(A129202(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2014

Table[Numerator[(n + 1)/2] Numerator[(n + 1)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 18 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1}, {1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54}, 60] (* Harvey P. Dale, Nov 20 2016 *)

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

a(n) = A129196(n)/(n+1).
(1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt = (a(n)*Pi^2-A129203(n))/A129196(n), i=sqrt(-1).
a(n) = ( Numerator of (n+1)/2 ) * ( Numerator of (n+1)/3 ) = A026741(n+1) * A051176(n+1). - Wesley Ivan Hurt, Jul 18 2014
G.f.: -(x^16 +2*x^15 +3*x^14 +8*x^13 +25*x^12 +6*x^11 +46*x^10 +26*x^9 +18*x^8 +26*x^7 +46*x^6 +6*x^5 +25*x^4 +8*x^3 +3*x^2 +2*x +1) / ((x -1)^3*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^3). - Colin Barker, Jul 18 2014
a(n+18) = 3*a(n+12)-3*a(n+6)+a(n). - Robert Israel, Jul 18 2014
a(n) = 2*(n+1)^2 * (7-4*cos(2*Pi*(n+1)/3)) / (9*(3-(-1)^n)). - Vaclav Kotesovec, Jul 20 2014
From Peter Bala, Feb 27 2019: (Start)
The following remarks assume an offset of 1.
a(n) = n^2/gcd(n,6) = n*A060789(n).
a(n) = n^2/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6. Thus a(n) is a quasi-polynomial in n:
a(6*n+1) = (6*n + 1)^2;
a(6*n+2) = 2*(3*n + 1)^2;
a(6*n+3) = 3*(2*n + 1)^2;
a(6*n+4) = 2*(3*n + 2)^2;
a(6*n+5) = (6*n + 5)^2;
a(6*n)   = 6*n^2.
O.g.f.: F(x) - 2*F(x^2) - 6*F(x^3) + 12*F(x^6), where F(x) = x*(1 + x)/(1 - x)^3 is the generating function for the squares. (End)
Sum_{n>=0} 1/a(n) = 55*Pi^2/216. - Amiram Eldar, Sep 27 2022

More terms from Wesley Ivan Hurt, Jul 18 2014

A227140 a(n) = n/gcd(n,2^5), n >= 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 2, 65, 33, 67, 17, 69, 35

Offset: 0

Wolfdieter Lang, Jul 04 2013

H(n,4) = 2*n*4/(n+4) is the harmonic mean of n and 4. For n >= 4 the denominator of H(n,4) is (n+4)/gcd(8*n,n+4) = (n+4)/gcd(n+4,32). a(n+8) = A227042(n+4,4), n >= 0. The numerator of H(n,4) is given in A227107. Thus a(n) is related to denominator of the harmonic mean H(n-4, 4).
Note the difference from A000265(n) (odd part of n) = n/gcd(n,2^n), n >= 1, which differs for the first time for n = 64. a(64) = 2, not 1.
A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000265, A227042, A227107, A106617, A276234.

List([0..80], n-> n/Gcd(n, 2^5)); # G. C. Greubel, Feb 27 2019

[n/GCD(n, 2^5): n in [0..80]]; // G. C. Greubel, Feb 27 2019

seq(n/igcd(n,32),n=0..70); # Muniru A Asiru, Feb 28 2019

With[{c=2^5},Table[n/GCD[n,c],{n,0,70}]] (* Harvey P. Dale, Feb 16 2018 *)

a(n)=n/gcd(n, 2^5); \\ Andrew Howroyd, Jul 23 2018

[n/gcd(n,2^5) for n in (0..80)] # G. C. Greubel, Feb 27 2019

a(n) = n/gcd(n, 2^5).
a(n) = denominator(8*(n-4)/n), n >= 0 (with denominator(infinity) = 0).
From Peter Bala, Feb 27 2019: (Start)
a(n) = numerator(n/(n + 32)).
O.g.f.: F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16) - F(x^32), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
From Bernard Schott, Mar 02 2019: (Start)
a(n) = 1 iff n is 1, 2, 4, 8, 16, 32 and a(2^n) = 2^(n-5) for n >= 5.
a(n) = n iff n is odd (A005408). (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^(e-min(e,5)), and a(p^e) = p^e for p > 2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/2^(2*s) - 1/2^(3*s) - 1/2^(4*s) - 1/2^(5*s)).
Sum_{k=1..n} a(k) ~ (683/2048) * n^2. (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A306368 a(n) = numerator of (n + 3)(n + 4)/((n + 1)(n + 2)).

Original entry on oeis.org

6, 10, 5, 21, 28, 12, 45, 55, 22, 78, 91, 35, 120, 136, 51, 171, 190, 70, 231, 253, 92, 300, 325, 117, 378, 406, 145, 465, 496, 176, 561, 595, 210, 666, 703, 247, 780, 820, 287, 903, 946, 330, 1035, 1081, 376, 1176, 1225, 425, 1326, 1378, 477, 1485, 1540, 532, 1653, 1711, 590

Offset: 0

Peter Bala, Feb 14 2019

If P(x) and Q(x) are coprime integral polynomials such that Q(n) > 0 for n >= 0 then the sequence of numerators of the rational numbers P(n)/Q(n) for n >= 0 and the sequence of denominators of P(n)/Q(n) for n >= 0 are both quasi-polynomial in n. In fact, there exists a purely periodic sequence b(n) such that numerator(P(n)/Q(n)) = P(n)/b(n) and denominator(P(n)/Q(n)) = Q(n)/b(n). Here we take P(n) = (n + 3)*(n + 4) and Q(n) = (n + 1)*(n + 2).

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Peter Bala, A note on the sequence of numerators of a rational function .
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A000326, A060544, A081266, A306367.

List([0..100],n->NumeratorRat((n+3)*(n+4)/((n+1)*(n+2)))); # Muniru A Asiru, Feb 25 2019

seq((n + 3)*(n + 4)/gcd((n + 3)*(n + 4), (n + 1)*(n + 2)), n = 0..100);

Table[((n+3)(n+4))/((n+1)(n+2)),{n,0,60}]//Numerator (* or *) LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{6,10,5,21,28,12,45,55,22},60] (* Harvey P. Dale, Mar 28 2020 *)

a(n) = numerator((n + 3)*(n + 4)/((n + 1)*(n + 2))); \\ Michel Marcus, Feb 26 2019

O.g.f.: (x^8 + x^7 - 3*x^5 - 2*x^4 + 3*x^3 + 5*x^2 + 10*x + 6)/((1 - x)^3*(x^2 + x + 1)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n >= 9.
a(n) = (n + 3)*(n + 4)/b(n), where (b(n))n>=0 is the purely periodic sequence [2, 2, 6, 2, 2, 6, ...] with period 3.
a(n) = (n + 3)*(n + 4)/gcd((n + 3)*(n + 4), (n + 1)*(n + 2)).
a(3*n)   = (3*n + 3)*(3*n + 4)/2 = A081266(n+1).
a(3*n+1) = (3*n + 4)*(3*n + 5)/2 = A060544(n+2).
a(3*n+2) = (n + 2)*(3*n + 5)/2   = A000326(n+2).
Sum_{n>=0} 1/a(n) = 2*log(3) - 2*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022

A027626 Denominator of n(n+5)/((n+2)(n+3)).

Original entry on oeis.org

1, 2, 10, 5, 7, 28, 12, 15, 55, 22, 26, 91, 35, 40, 136, 51, 57, 190, 70, 77, 253, 92, 100, 325, 117, 126, 406, 145, 155, 496, 176, 187, 595, 210, 222, 703, 247, 260, 820, 287, 301, 946, 330, 345, 1081, 376, 392, 1225, 425, 442, 1378, 477, 495, 1540, 532, 551

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A000292, A027625, A061347, A107711, A234041.

[Denominator(n*(n+5)/((n+2)*(n+3))): n in [0..60]]; // Vincenzo Librandi, Mar 04 2014

CoefficientList[Series[(1+2*x+10*x^2+2*x^3+x^4-2*x^5+x^8)/(1-x^3)^3, {x,0,50}], x] (* Vincenzo Librandi, Mar 04 2014 *)

a(n) = numerator((n+2)*(n+3)/6); \\ Altug Alkan, Apr 18 2018

[numerator(binomial(n+3,2)/3) for n in (0..60)] # G. C. Greubel, Aug 04 2022

a(n) = GCD of n-th and (n+1)st tetrahedral numbers (A000292). - Ross La Haye, Sep 13 2003
G.f.: (1 +2*x +10*x^2 +2*x^3 +x^4 -2*x^5 +x^8)/(1-x^3)^3.
a(n) = A234041(n+1) = A107711(n+4,3) = C(n+3,2)*gcd(n+4,3)/3 for n >= 0. See the o.g.f. of A234041. - Wolfdieter Lang, Feb 26 2014
a(n) = numerator of (n+2)*(n+3)/6. - Altug Alkan, Apr 18 2018
Sum_{n>=0} 1/a(n) = 5 - 4*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022
a(n) = (n + 2)*(n + 3)*(5 - 2*A061347(n+1))/18. - Stefano Spezia, Oct 16 2023
a(n) is quasi-polynomial in n: a(3*n) = (n+1)*(3*n+2)/2 = A000326(n+1); a(3*n+1) = (n+1)*(3*n+4)/2 = A005449(n+1); a(3*n+2) = (3*n+4)*(3*n+5)/2 = A060544(n+2). - Peter Bala, Nov 20 2024

More terms from Vincenzo Librandi, Mar 04 2014

A276234 a(n) = n/gcd(n, 256).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 33, 67, 17, 69, 35, 71

Offset: 1

Artur Jasinski, Aug 24 2016

a(n) first differs from A000265(n) at n = 512. - Andrew Howroyd, Jul 23 2018

A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, order 512.

Cf. A276233 (numerators), A227140, A000265, A106617.

List([1..80], n-> n/Gcd(n, 2^8)); # G. C. Greubel, Feb 27 2019

[n/GCD(n, 2^8): n in [1..80]]; // G. C. Greubel, Feb 27 2019

seq(n/igcd(n,256), n=1..100); # Robert Israel, Aug 26 2016

Table[n/GCD[n, 2^8], {n,1,80}] (* G. C. Greubel, Feb 27 2019 *)

a(n)=n/gcd(n,256) \\ Charles R Greathouse IV, Aug 26 2016

[n/gcd(n, 2^8) for n in (1..80)] # G. C. Greubel, Feb 27 2019

a(2k-1) = 2k-1.
G.f.: (x+x^3)/(1-x^2)^2 +(x^2+x^6)/(1-x^4)^2 +(x^4+x^12)/(1-x^8)^2 +(x^8+x^24)/(1-x^16)^2 +(x^16+x^48)/(1-x^32)^2 +(x^32+x^96)/(1-x^64)^2 +(x^64+x^192)/(1-x^128)^2 +(x^128+x^256+x^384)/(1-x^256)^2. - Robert Israel, Aug 26 2016
a(n) = 2*a(n-256) - a(n-512). - Charles R Greathouse IV, Aug 26 2016
From Peter Bala, Feb 27 2019: (Start)
a(n) = numerator(n/(n + 256)).
O.g.f.: F(x) - Sum_{k = 1..8} F(x^(2^k)), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
From Amiram Eldar, Nov 26 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/2^(2*s) - 1/2^(3*s) - 1/2^(4*s) - 1/2^(5*s) - 1/2^(6*s) - 1/2^(7*s) - 1/2^(8*s)).
Multiplicative with a(2^e) = 2^(e-min(e,8)), and a(p^e) = p^e for p > 2.
Sum_{k=1..n} a(k) ~ (43691/131072) * n^2. (End)

Keyword:mult added and terms a(51) and beyond from Andrew Howroyd, Jul 23 2018

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A129202 Denominator of 3*(3+(-1)^n) / (n+1)^2.

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A227140 a(n) = n/gcd(n,2^5), n >= 0.

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A306368 a(n) = numerator of (n + 3)*(n + 4)/((n + 1)*(n + 2)).

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A027626 Denominator of n*(n+5)/((n+2)*(n+3)).

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A276234 a(n) = n/gcd(n, 256).

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A306368 a(n) = numerator of (n + 3)(n + 4)/((n + 1)(n + 2)).

A027626 Denominator of n(n+5)/((n+2)(n+3)).