A138190 -id:A138190 - cp's OEIS frontend

A138191 Denominator of (n-1)n(n+1)/12.

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Eric W. Weisstein, Mar 04 2008

Proof of 4-periodicity follows from evaluating (n+3)(n+4)(n+5)/12, subtracting (n-1)n(n+1)/12 and getting n^2+4n+5 which is an integer. - R. J. Mathar, Mar 07 2008

			0, 1/2, 2, 5, 10, 35/2, 28, 42, 60, 165/2, 110, 143, 182, ...

Eric Weisstein's World of Mathematics, Kirchhoff Index.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A000292, A107453, A107453, A138190.

Table[(n^3-n)/12,{n,120}]//Denominator (* or *) PadRight[{},120,{1,2,1,1}] (* Harvey P. Dale, Apr 15 2019 *)

def A138191(n): return (1,1,2,1)[n&3] # Chai Wah Wu, Apr 25 2024

From R. J. Mathar, Mar 07 2008: (Start)
a(n) = 1 + (A000292(n-1) mod 2) = a(n-4).
O.g.f.: -1-5/(4(x-1))+1/(4(x+1))-1/(2(x^2+1)). (End)
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and 1 otherwise.
Dirichlet g.f.: zeta(s)*(1+1/2^s-1/4^s).
Sum_{k=1..n} a(k) ~ (5/4)*n. (End)

A208950 a(4n) = n(16n^2-1)/3, a(2n+1) = n(n+1)(2n+1)/6, a(4n+2) = (4n+1)(4n+2)(4*n+3)/6.

Original entry on oeis.org

0, 0, 1, 1, 5, 5, 35, 14, 42, 30, 165, 55, 143, 91, 455, 140, 340, 204, 969, 285, 665, 385, 1771, 506, 1150, 650, 2925, 819, 1827, 1015, 4495, 1240, 2728, 1496, 6545, 1785, 3885, 2109, 9139, 2470, 5330, 2870, 12341, 3311, 7095, 3795, 16215, 4324

Offset: 0

Paul Curtz, Mar 03 2012

a(n+2) is divisible by A060819(floor(n/3)).
a(n) is divisible by A176672(floor(n/3)).
Denominator of a(n)/n is of period 24: 1,1,3,4,1,6,1,4,3,1,1,12,1,2,3,4,1,3,1,4,3,2,1,12 (two successive palindromes).
This is the fifth column of the triangle A107711, hence the formula involving gcd(n+2,4) given below follows. - Wolfdieter Lang, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A107711 (fifth column).

Cf. A000034, A000292, A002415, A051724, A060819, A061037, A107711, A138190, A145979, A176672, A176895.

[Binomial(n+1,3)*GCD(n+2,4)/4: n in [0..50]]; // G. C. Greubel, Sep 20 2018

CoefficientList[Series[(x^2 + x^3 + 5 x^4 + 5 x^5 + 31 x^6 + 10 x^7 + 22 x^8 + 10 x^9 + 31 x^10 + 5 x^11 + 5 x^12 + x^13 + x^14)/((1 - x)^4 (1 + x)^4 (1 + 4 x^2 + 6 x^4 + 4 x^6 + x^8)), {x, 0, 47}], x] (* Bruno Berselli, Mar 11 2012 *)

A208950(n) := block(
        [a,npr] ,
        if equal(mod(n,4), 0) then (
                a : n/12*(n^2-1)
        ) else if equal(mod(n,2),0) then (
                a : (n-1)*n*(n+1)/6
        ) else (
                npr : (n-1)/2,
                a : npr*(npr+1)*n/6
        ) ,
        return(a)
)$ /* R. J. Mathar, Mar 10 2012 */

vector(50, n, n--; binomial(n+1,3)*gcd(n+2,4)/4) \\ G. C. Greubel, Sep 20 2018

a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n+1) = A002415(n+1)/A145979(n-1).
a(n) = A051724(n-1) * A051724(n) * A051724(n+1).
a(n) = A060819(n-1) * A060819(n) * A060819(n+1) / 3.
a(n) * a(n+4) = A061037(n+1) * A061037(n+2) * A061037(n+3) / 9.
a(n) = A138190(n)/A000034(n) for n > 0.
a(n) = A000292(n-1)/A176895(n+2) for n > 0.
a(n)/a(n+4) = n*(n^2-1)/((n+3)*(n+4)*(n+5)).
a(n)/a(n+12) = (n-1)*n*(n+1)/((n+11)*(n+12)*(n+13)).
G.f.: (x^2 + x^3 + 5*x^4 + 5*x^5 + 31*x^6 + 10*x^7 + 22*x^8 + 10*x^9 + 31*x^10 + 5*x^11 + 5*x^12 + x^13 + x^14) / ((1-x)^4*(1+x)^4*(1 + 4*x^2 + 6*x^4 + 4*x^6 + x^8)). - R. J. Mathar, Mar 10 2012
From Wolfdieter Lang, Feb 24 2014: (Start)
G.f.: (1 + x^12 + x*(1+x^10) + 5*x^2*(1+x^8) + 5*x^3*(1+x^7) + 31*x^4*(1+x^4) + 10*x^5*(1+x^2) + 22*x^6)/(1-x^4)^4. This is the preceding g.f. rewritten.
a(n) = binomial(n+1,3)*gcd(n+2,4)/4, n >= 0. From the g.f., see a comment above on A107711. (End)
a(n) = (n*(n-1)*((n+1)*(4+2*(-1)^n + (1+(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))))/48. - Luce ETIENNE, Jan 01 2015
Sum_{n>=2} 1/a(n) = 12 - 27*log(2)/2. - Amiram Eldar, Aug 12 2022

A276670 Numerator of (n-1)n(n+1)/4.

Original entry on oeis.org

0, 0, 3, 6, 15, 30, 105, 84, 126, 180, 495, 330, 429, 546, 1365, 840, 1020, 1224, 2907, 1710, 1995, 2310, 5313, 3036, 3450, 3900, 8775, 4914, 5481, 6090, 13485, 7440, 8184, 8976, 19635, 10710, 11655, 12654, 27417, 14820, 15990, 17220, 37023, 19866, 21285

Offset: 0

Paul Curtz, Oct 05 2016

Consider the sequence [2/(n+1), autosequence of the second kind] (see A003506), and its successive differences:
    2,    1,  2/3,   1/2,   2/5,   1/3,   2/7,   1/4,   2/9, ... (see A026741)
   -1, -1/3, -1/6, -1/10, -1/15, -1/21, -1/28, -1/36, -1/45, ... (see A000217)
  2/3,  1/6, 1/15,  1/30, 2/105,  1/84, 1/126, 1/180, 2/495, ...
  ...
Each fraction in the third row is essentially the reciprocal of (n-1)*n*(n+1)/4 (3/2, 6, 15, 30, 105/2, ... ).
The numbers (= 3*A138190) are divisible by
1) -1, 1, 1, 1, 3, 2, 5, 3, 7, ... hence f(n) = 0, 0, 3, 6, 5, 15, 21, 28, 18, ...
2)  1, 1, 3, 3, 5, 5, 7, 7, 9, ... hence g(n) = 0, 0, 1, 2, 3, 6, 15, 12, 14, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A000217, A003506, A007531, A027641, A138190, A109613, A236398 (denominators).

seq(numer((n^3-n)/4), n=0..100); # Robert Israel, Oct 05 2016

f[n_] := Numerator[(n - 1) n (n + 1)/4]; Array[f, 40, 0] (* Robert G. Wilson v, Oct 05 2016 *)

concat(vector(2), Vec(3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4) + O(x^30))) \\ Colin Barker, Oct 09 2016

a(n) = 3*A138190(n), for n>=1.
a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n) = A007531(n+1)/2 if n == 2 (mod 4), otherwise a(n) = A007531(n+1)/4. - Robert Israel, Oct 05 2016
G.f.: 3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4). - Colin Barker, Oct 09 2016
Sum_{n>=2} 1/a(n) = 1 - log(2)/2. - Amiram Eldar, Aug 13 2022

More terms from Robert G. Wilson v, Oct 05 2016

cp's OEIS Frontend

A138191 Denominator of (n-1)n(n+1)/12.

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A208950 a(4n) = n(16n^2-1)/3, a(2n+1) = n(n+1)(2n+1)/6, a(4n+2) = (4n+1)(4n+2)(4*n+3)/6.

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A276670 Numerator of (n-1)n(n+1)/4.

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cp's OEIS Frontend

A138191 Denominator of (n-1)*n*(n+1)/12.

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A208950 a(4*n) = n*(16*n^2-1)/3, a(2*n+1) = n*(n+1)*(2*n+1)/6, a(4*n+2) = (4*n+1)*(4*n+2)*(4*n+3)/6.

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A276670 Numerator of (n-1)*n*(n+1)/4.

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A138191 Denominator of (n-1)n(n+1)/12.

A208950 a(4n) = n(16n^2-1)/3, a(2n+1) = n(n+1)(2n+1)/6, a(4n+2) = (4n+1)(4n+2)(4*n+3)/6.

A276670 Numerator of (n-1)n(n+1)/4.