A109007 -id:A109007 - cp's OEIS frontend

A130334 Smallest m>n such that the m-th and n-th triangular numbers are coprime.

2, 4, 10, 6, 7, 10, 9, 10, 13, 12, 13, 22, 15, 16, 22, 18, 19, 22, 21, 22, 25, 24, 25, 37, 27, 28, 37, 30, 31, 37, 33, 34, 37, 36, 37, 46, 39, 40, 46, 42, 43, 46, 45, 46, 52, 48, 49, 58, 51, 52, 58, 54, 55, 58, 57, 58, 61, 60, 61, 73, 63, 64, 73, 66, 67, 70, 69, 70, 73, 72, 73

Offset: 1

Reinhard Zumkeller, May 28 2007

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Eric Weisstein's World of Mathematics, Relatively Prime

Cf. A000217, A026741, A109007, A130335.

from math import gcd
def A130334(n):
    k, Tn, Tm = n+1, n*(n+1)//2, (n+1)*(n+2)//2
    while gcd(Tn,Tm) != 1:
        k += 1
        Tm += k
    return k # Chai Wah Wu, Sep 16 2021

a(n) > n+1 for n>1; a(n) > n+2 for n with n mod 3 = 0;

a(n) = n + A130335(n).

A130724 a(n) = lcm(n,3) / gcd(n,3).

Original entry on oeis.org

0, 3, 6, 1, 12, 15, 2, 21, 24, 3, 30, 33, 4, 39, 42, 5, 48, 51, 6, 57, 60, 7, 66, 69, 8, 75, 78, 9, 84, 87, 10, 93, 96, 11, 102, 105, 12, 111, 114, 13, 120, 123, 14, 129, 132, 15, 138, 141, 16, 147, 150, 17, 156, 159, 18, 165, 168, 19, 174, 177, 20, 183, 186

Offset: 0

W. Neville Holmes, Jul 04 2007

			a(7) = 21 because lcm(3,7) = 21, gcd(3,7) = 1 and 21/1 = 21.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A109007, A109044.

[Lcm(n,3)/Gcd(n,3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2016

A130724:=n->lcm(n,3)/gcd(n,3): seq(A130724(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2016

LCM[3,#]/GCD[3,#]&/@Range[0,70] (* Harvey P. Dale, May 16 2013 *)

a(n) = A109044(n) / A109007(n).
From Wesley Ivan Hurt, Jul 24 2016: (Start)
G.f.: x*(3 + 6*x + x^2 + 6*x^3 + 3*x^4)/(x^3 - 1)^2.
a(n) = 2*a(n-3) - a(n-6) for n>5.
a(n) = 27*n/(5 + 4*cos(2*n*Pi/3))^2.
If n mod 3 = 0, then n/3, else 3*n.
a(n) = lcm(numerator(n/3), denominator(n/3)). (End)
Sum_{k=1..n} a(k) ~ (19/18)*n^2. - Amiram Eldar, Oct 07 2023

Corrected and extended by Harvey P. Dale, May 16 2013

A234041 a(n) = binomial(n+2,2)*gcd(n,3)/3, n >= 0.

Original entry on oeis.org

1, 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

Offset: 0

Wolfdieter Lang, Feb 24 2014

Apart from the first term, this is the same as A027626. - Bruno Berselli, Feb 24 2014

This is the sequence of the fourth column of the triangle A107711.

			a(6) = binomial(8,2) = 28 (example for n == 0 (mod 3)),
a(7) = binomial(9,2)/3 = 3*4 = 12 (example for n == 1 (mod 3)),
a(8) = binomial(10,2)/3 = 5*3 = 15 (example for n == 2 (mod 3)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A027626, A107711, A026741 (third column of A107711), A109007 (gcd(n,3)).

Table[Binomial[n + 2, 2] GCD[n + 3, 3]/3, {n, 0, 60}] (* Bruno Berselli, Feb 24 2014 *)
CoefficientList[Series[(1 + x + 2 x^2 + 7 x^3 + 2 x^4 + x^5 + x^6)/(1 - x^3)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 26 2014 *)

a(n) = numerator((n+1)*(n+2)/6); \\ Altug Alkan, Apr 19 2018

G.f.: (1+x+2*x^2+7*x^3+2*x^4+x^5+x^6)/(1-x^3)^3.
a(n) = A107711(n+3,3) for n >= 0.
a(n) = (2+(-1)^(n+floor((n+1)/3)))*(n+1)*(n+2)/6. - Bruno Berselli, Feb 24 2014
a(n) is the numerator of (n+1)*(n+2)/6. - Altug Alkan, Apr 19 2018
Sum_{n>=0} 1/a(n) = 6 - 4*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022

A132951 Period 6: repeat [1, 3, 1, -1, -3, -1].

Original entry on oeis.org

1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1

Offset: 0

Paul Curtz, Nov 22 2007

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

Cf. A109007, A130151, A131531.

&cat [[1, 3, 1, -1, -3, -1]: n in [0..20]]; // Wesley Ivan Hurt, Nov 18 2022

PadRight[{},120,{1,3,1,-1,-3,-1}] (* Harvey P. Dale, Feb 26 2024 *)

a(n)=[1,3,1,-1,-3,-1][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 3*a(n-1)-a(n-3)+3*a(n-4).
O.g.f.: (1+3*x+x^2)/((x+1)*(x^2-x+1)) = -(1/3)/(x+1)+(1/3)*(4*x+4)/(x^2-x+1). - R. J. Mathar, Nov 28 2007
a(n) = -(1/3)*(-1)^n+(4/3)*cos(Pi*n/3)+(4*3^0.5/3)*sin(Pi*n/3). - Richard Choulet, Jan 02 2008
a(n) = a(n-6) = A131531(n+3)+A131531(n+1)+3*A131531(n+2). - R. J. Mathar, Apr 04 2008
a(n) = A109007(n+2) * A130151(n). - Wesley Ivan Hurt, Jun 22 2013

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar.

A216913 a(n) = Gauss_primorial(3n, 3) / Gauss_primorial(3n, 3*n).

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 7, 2, 1, 10, 11, 2, 13, 14, 5, 2, 17, 2, 19, 10, 7, 22, 23, 2, 5, 26, 1, 14, 29, 10, 31, 2, 11, 34, 35, 2, 37, 38, 13, 10, 41, 14, 43, 22, 5, 46, 47, 2, 7, 10, 17, 26, 53, 2, 55, 14, 19, 58, 59, 10, 61, 62, 7, 2, 65, 22, 67, 34, 23, 70, 71

Offset: 1

Peter Luschny, Oct 02 2012

The term Gauss primorial was introduced in A216914 and denotes the restriction of the Gauss factorial N_n! (see A216919) to prime factors.

Multiplicative because both A007947 and A109007 are. - Andrew Howroyd, Aug 02 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A007947, A065463, A109007, A204455, A216914, A216919.

[&+[EulerPhi(d)*MoebiusMu(3*d)^2:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Oct 19 2019

Table[n/Sum[Floor[Cos[Pi k^(3 n)/(3 n)]^2], {k, 3 n}], {n, 71}] (* Michael De Vlieger, May 24 2016 *)
a[n_] := Times @@ (First /@ FactorInteger[n])/GCD[n, 3]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

a(n)={factorback(factor(n)[, 1])/gcd(3,n)} \\ Andrew Howroyd, Aug 02 2018

def Gauss_primorial(N, n):
    return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
def A216913(n): return Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n)
[A216913(n) for n in (1..80)]

a(n) = n/Sum_{k=1..3n} floor(cos^2(Pi*k^(3n)/(3n))). - Anthony Browne, May 24 2016
a(n) = A007947(n)/A109007(n). - Andrew Howroyd, Aug 02 2018
a(n) = Sum_{d|n} phi(d)*mu(3d)^2. - Ridouane Oudra, Oct 19 2019
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(3^e) = 1, and a(p^e) = p for p != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (9/22) * Product_{p prime} (1 - 1/(p*(p+1))) = (9/22) * A065463 = 0.2881809... . (End)

A194130 a(n) = n!/gcd(n,3).

Original entry on oeis.org

1, 2, 2, 24, 120, 240, 5040, 40320, 120960, 3628800, 39916800, 159667200, 6227020800, 87178291200, 435891456000, 20922789888000, 355687428096000, 2134124568576000, 121645100408832000, 2432902008176640000, 17030314057236480000, 1124000727777607680000

Offset: 1

Paul Curtz, Aug 16 2011

nonn

Cf. A109007, A144396, A005563, A051925.

A194130 := proc(n)
        n!/igcd(n,3) ;
end proc:
seq(A194130(n),n=1..30) ;

Table[n!/GCD[n,3],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)

Definition and offset corrected by R. J. Mathar, Aug 18 2011

A203648 a(n) = (1/4) * period of repeating sequence {S(j) mod 2n}, where S(j) is the sum of the first j squares.

Original entry on oeis.org

1, 2, 9, 4, 5, 18, 7, 8, 27, 10, 11, 36, 13, 14, 45, 16, 17, 54, 19, 20, 63, 22, 23, 72, 25, 26, 81, 28, 29, 90, 31, 32, 99, 34, 35, 108, 37, 38, 117, 40, 41, 126, 43, 44, 135, 46, 47, 144, 49, 50, 153, 52, 53, 162, 55, 56, 171, 58, 59, 180, 61, 62, 189, 64, 65

Offset: 1

Gary Detlefs, Jan 04 2012

This sequence lists the periods of the sum of the first n squares mod 2*n. In most cases, (Sum_{k=1..n} k^(2*j)) mod 2*n will produce the same sequence. The repeating sequences appear to always end in 2 zeros.
(Sum_{k=1..n} k^j) mod 2 has period 4 repeating [1,1,0,0] for any j.
It appears that a(n) is the number of n-colorings of the trefoil knot. - Tsuyoshi Miezaki, May 01 2022

			G.f. = x + 2*x^2 + 9*x^3 + 4*x^4 + 5*x^5 + 18*x^6 + 7*x^7 + 8*x^8 + 27*x^9 + ...
(Sum_{k=1..n} k^2) mod 4 has period 8 repeating [1,1,2,2,3,3,0,0] so a(2) = 2.

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

Cf. A008585 (3*n), A109007.

[n*(1+2*Floor(((n+2) mod 3)/2)): n in [1..60]]; // Vincenzo Librandi, Mar 19 2012

seq(n*(1+floor(((n+2) mod 3)/2))), n= 1..57);

CoefficientList[Series[(1+2*x+9*x^2+2*x^3+x^4)/((1-x)^2*(1+x+x^2)^2),{x,0,60}],x] (* Vincenzo Librandi, Mar 19 2012 *)
Table[n (1 + 2 Floor[Mod[n + 2, 3]/2]), {n, 57}] (* Michael De Vlieger, Jan 14 2017 *)
LinearRecurrence[{0,0,2,0,0,-1},{1,2,9,4,5,18},70] (* Harvey P. Dale, Sep 13 2024 *)

{a(n) = if( n%3, n, 3*n)}; /* Michael Somos, Jan 18 2017 */

a(n) = 3*n if n mod 3 = 0, otherwise n.
a(n) = n*(1 + 2*floor(((n+2) mod 3)/2)).
From Bruno Berselli, Jan 04 2012: (Start)
G.f.: x*(1 + 2*x + 9*x^2 + 2*x^3 + x^4)/((1-x)^2*(1 + x + x^2)^2).
a(n) = 2*n + 2*n*((-1)^(-2*n/3) + (-1)^(2*n/3)-1/2)/3.
a(n) = -a(-n) = 2*a(n-3) - a(n-6). (End)
a(n) = numerator(3n/((3 + 2*(-1 + n))*(1 + n))). - Andres Cicuttin, Jan 12 2017
a(n) is multiplicative with a(p^e) = p^(e+1) if p = 3, a(p^e) = p^e otherwise. - Michael Somos, Jan 18 2017
a(n) = n*(5 + 4*cos((2*Pi*n)/3)) / 3. - Colin Barker, Mar 06 2017
From Amiram Eldar, Dec 27 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(3^(-s)*(6 + 3^s)).
Sum_{k=1..n} a(k) ~ (5/6) * n^2. (End)
a(n) = n*A109007(n). - Lechoslaw Ratajczak, Aug 16 2023
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*log(2)/9. - Amiram Eldar, Aug 21 2023

A244840 Denominators of the triangle T(n,k) = (n*(n+1)/2+k+1)/(k+1) for n >= k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 5, 2, 1, 1, 1, 3, 1, 5, 3, 1, 2, 1, 1, 1, 1, 5, 1, 7, 2, 1, 1, 2, 1, 4, 1, 2, 7, 8, 1, 2, 1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 1, 1, 1, 1, 2, 5, 1, 7, 4, 3, 5, 1, 2

Offset: 0

Paul Curtz, Jul 07 2014

Numerators: A244734(n,k).

See A244734 for the first entries of the rational triangle T(n,k).

			T(0,0) = 1/1, T(1,0) = 2/1, T(1,1) = 3/2,... .
The triangle a(n,k) begins:
n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...
0:   1
1:   1 2
2:   1 2 1
3:   1 1 1 2
4:   1 1 3 2 1
5:   1 2 1 4 1 2
6:   1 2 1 4 5 2 1
7:   1 1 3 1 5 3 1 2
8:   1 1 1 1 5 1 7 2 1
9:   1 2 1 4 1 2 7 8 1  2
10:  1 2 3 4 1 6 7 8 9  2  1
11:  1 1 1 2 5 1 7 4 3  5  1  2
12:  1 1 1 2 5 1 7 4 3  5 11  2 1
13:  1 2 3 4 5 6 1 8 9 10 11 12 1   2
14:  1 2 1 4 1 2 1 8 3  2 11  4 13  2  1
15:  1 1 1 1 1 1 7 1 3  1 11  1 13  7  1  2
16:  1 1 3 1 5 3 7 1 9  5 11  3 13  7 15  2  1
17:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16  1  2
18:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16 17  2  1
19:  1 1 3 2 1 3 7 4 9  1 11  6 13  7  3  8 17  9  1  2
20:  1 1 1 2 1 1 1 4 3  1 11  2 13  1  1  8 17  3 19  2  1
n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...
.. reformatted - _Wolfdieter Lang_, Jul 28 2014 .
The second column is of period 4: repeat 2, 2, 1, 1. From A014695 or A130658.
The third column is of period 3: repeat 1, 1, 3. From A109007.
The fourth column is of period 8: repeat 2, 2, 4, 4, 1, 1, 4, 4.
The fifth column is of period 5: repeat 1, 1, 5, 5, 5.
The sixth column is of period 12: repeat 2, 2, 3, 1, 2, 6, 1, 1, 6, 2, 1, 3 .
The seventh column is of period 7: repeat 1, 1, 7, 7, 7, 7, 7.
Hence the positive terms of A022998.
Main diagonal: A000034(n).
Alternate main and second diagonal: A130658(n).
Common denominator by row: 1, 2, 2, 2, 6, 4, 20, 30, 70, ... .

Cf. A014695, A130658, A109007, A002260, A022998, A244734, A000034.

Table[(n*(n+1)/2+k+1)/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2014 *)

a(n,k) = denominator((n*(n+1)/2 + k + 1)/(k+1)) for n >= k >= 0.

Editse: Name reformulated, comment with T(n,k) reference added. - Wolfdieter Lang, Jul 28 2014

A257106 Denominators of the inverse binomial transform of the Bernoulli numbers with B(1)=2/3.

Original entry on oeis.org

1, 3, 6, 2, 10, 6, 42, 6, 30, 2, 22, 6, 2730, 6, 6, 2, 170, 6, 798, 6, 330, 2, 46, 6, 2730, 6, 6, 2, 290, 6, 14322, 6, 510, 2, 2, 6, 1919190, 6, 6, 2, 4510, 6, 1806, 6, 690, 2, 94, 6, 46410, 6, 66, 2, 530, 6, 798, 6, 870, 2, 118, 6, 56786730, 6, 6, 2, 170, 6

Offset: 0

Paul Curtz, Apr 23 2015

nonn

Difference table of Bernoulli numbers with B(1)=2/3:
1,       2/3,    1/6,      0, -1/30,     0,  1/42, 0, ...
-1/3,   -1/2,   -1/6,  -1/30,  1/30,  1/42, -1/42, ...
-1/6,    1/3,   2/15,   1/15, -1/105, -1/21, ...
1/2,    -1/5,  -1/15, -8/105, -4/105, ...
-7/10,  2/15, -1/105,  4/105, ...
5/6,    -1/7,   1/21, ...
-41/42, 2/15, ...
7/6, ...
...
First column: 1, -1/3, -1/6, 1/2, -7/10, 5/6, -41/42, 7/6, -41/30, 3/2, -35/22, 11/6, ... . a(n) is the n-th term of the denominators.
Antidiagonal sums: 1, 1/3, -1/2, 2/3, -5/6, 1, -7/6, 4/3, -3/2, 5/3, -11/6, 2, ... . See A060789(n).
a(2n+2)/a(2n+1) = 2, 5, 7, 5, 11, 455, ... .
By definition, for B(1) = b, the inverse binomial transform is
Bi(b) =       1,  -1 + b, 7/6 - 2*b, -3/2 + 3*b, 59/30 + 4*b, ...
      = A176328(n)/A176591(n) - (-1)^n *n*b.
With Bic(b) = 0, -1/2 + b, 1 - 2*b,  -3/2 + 3*b,   2 + 4*b, ...
            = (-1)^n *(A001477(n)/2 - n*b),
Bi(b) = (-1)^n *(A164555(n)/A027642(n) + A001477(n)/2 - n*b) =
      = A027641(n)/A027642(n) + Bic(b) .

			a(0) = 1-0, a(1) = -1/2 +1/6 = -1/3, a(2) = 1/6 -1/3 = -1/6, a(3) = 0 +1/2.

Cf. A256595, A027641/A027642(n), A164555(n)/A027642(n), A060789, A176328/A176591, A001477, A109007.

max = 66; B[1] = 2/3; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Denominator; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 11 2015 *)

def A257106_list(len, B1) :
    T = matrix(QQ, 2*len+1)
    for m in (0..2*len) :
        T[0, m] = bernoulli_polynomial(1, m) if m <> 1 else B1
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return [denominator(T[k, 0]) for k in (0..len-1)]
A257106_list(66, 2/3) # Peter Luschny, May 09 2015

Conjecture: a(2n+1) = 3 followed by period 3: repeat 2, 6, 6.
Conjecture: a(2n) = A002445(n)/(period 3: repeat 1, 1, 3).
a(n) = A027641(n)/A027642(n) - (-1)^n *n/6.

A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1

Offset: 0

Paul Curtz, Jan 14 2017

Successive sequences:
   0,   0,  0,    0, ...    = 0 * ( )
   4,  -3,  11,  -8, ...    = 1 * ( )
   1,   8,   3,  16, ...    = 1 * ( )                 A195161
  12,   0,  27,  -3, ...    = 3 * (4, 0, 9, -1, ...)
   4,  24,   8,  40, ...    = 4 * (1, 6, 2, 10, ...)  A064680
5;   28,   5,  51,   4, ...    = 1 * ( )
   9,  48,  15,  72, ...    = 3 * (3, 16, 5, 24, ...) A195161
  52,  12,  83,  13, ...    = 1 * ( )
  16,  80,  24, 112, ...    = 8 * (2, 10, 3, 14, ...) A064080
  84   21, 123,  24, ...    = 3 * (28, 7, 41, 8, ...)
 25, 120,  35, 160, ...    = 5 * (5, 24, 7, 32, ...) A195161

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1,0,2,0,1,0,-1,0,-1).

Cf. A064680, A144433 or A195161.

Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)

f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017

A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017
a(2*k)   = A022998(k).
a(2*k+1) = A109007(k-1).
a(3*k)   = interleave 3*k*(3 +(-1)^k)/2, 3.
a(3*k+1) = interleave 1, A166304(k).
a(3*k+2) = interleave A166138(k), 1.
a(4*k)   = 4*k.
a(4*k+1) = period 3: repeat [1, 1, 3].
a(4*k+2) = 1 + 2*k.
a(4*k+3) = period 3: repeat [3, 1, 1].
a(n+12) - a(n) = 6*A131743(n+3).
a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Corrected and extended by Michel Marcus, Jan 15 2017

cp's OEIS Frontend

A130334 Smallest m>n such that the m-th and n-th triangular numbers are coprime.

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A130724 a(n) = lcm(n,3) / gcd(n,3).

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A234041 a(n) = binomial(n+2,2)*gcd(n,3)/3, n >= 0.

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A132951 Period 6: repeat [1, 3, 1, -1, -3, -1].

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A216913 a(n) = Gauss_primorial(3*n, 3) / Gauss_primorial(3*n, 3*n).

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A194130 a(n) = n!/gcd(n,3).

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A203648 a(n) = (1/4) * period of repeating sequence {S(j) mod 2n}, where S(j) is the sum of the first j squares.

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A216913 a(n) = Gauss_primorial(3n, 3) / Gauss_primorial(3n, 3*n).