A176672 -id:A176672 - cp's OEIS frontend

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

0, 1, 1, 2, 5, 5, 7, 28, 3, 15, 55, 22, 13, 91, 35, 40, 34, 51, 57, 190, 35, 77, 253, 92, 25, 325, 117, 126, 203, 145, 155, 496, 44, 187, 595, 210, 111, 703, 247, 260, 205, 287, 301, 946, 165, 345, 1081, 376, 98, 1225, 425

Offset: 0

Paul Curtz, Aug 28 2011

Akiyama-Tanigawa algorithm from 1/n leads to Bernoulli A164555(n)/A027642(n):
1,   1/2,  1/3,  1/4,
1/2, 1/3,  1/4,  1/5,
1/6, 1/6, 3/20, 2/15,           =A026741(n+1)/A045896(n+1),
0,  1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495  =a(n)/b(n).

Cf. A193220 (denominators).

a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[3, k], {k, 0, 50}] // Numerator (* Jean-François Alcover, Sep 19 2012 *)

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

A100652 Denominator of 1 - Sum_{i=1..n} |Bernoulli(i)|.

Original entry on oeis.org

1, 2, 3, 3, 10, 10, 105, 105, 70, 70, 1155, 1155, 1430, 1430, 2145, 2145, 24310, 24310, 4849845, 4849845, 58786, 58786, 2028117, 2028117, 965770, 965770, 1448655, 1448655, 28007330, 28007330, 100280245065, 100280245065, 66853496710, 66853496710, 100280245065

Offset: 1

N. J. A. Sloane, Dec 05 2004

Contribution from Paul Curtz, Aug 07 2012 (Start):
Take a(0)=1. Then instead of the Akiyama-Tanigawa algorithm we create the extended (or prolonged) Akiyama-Tanigawa algorithm using A028310(n)=1,1,2,3,4,5,... instead of A000027(n)=1,2,3,4,5,.. .
Hence the array (A051714 with an additional column)
2,       1,   1/2,   1/3,     1/4,
1,     1/2,   1/3,   1/4,     1/5,
1/2,   1/6,   1/6,   3/20,   2/15,       A026741(n+1)/A045896(n+1)
1/3,     0,  1/30,   1/20,   2/35,       A194531(n)/A193220(n)
1/3, -1/30, -1/30, -3/140, -1/105.       A051722(n)/A051723(n).
a(n) is the denominator of the (first) column before the Akiyama-Tanigawa algorithm leading to the second Bernoulli numbers A164555(n)/A027642(n). See A176672(n).
(End)

			1, 1/2, 1/3, 1/3, 3/10, 3/10, 29/105, 29/105, 17/70, 17/70, 193/1155, 193/1155, -123/1430, -123/1430, -2687/2145, -2687/2145, -202863/24310, -202863/24310, -307072861/4849845, ... = A100651/A100652.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Denominator[1-(Accumulate[Abs[BernoulliB[Range[0,40]]]])] (* Harvey P. Dale, Jan 28 2013 *)

A178370 The trisection A178242(3n+2).

Original entry on oeis.org

7, 25, 26, 44, 133, 187, 125, 161, 403, 493, 296, 350, 817, 943, 539, 611, 1375, 1537, 854, 944, 2077, 2275, 1241, 1349, 2923, 3157, 1700, 1826, 3913, 4183, 2231, 2375, 5047, 5353, 2834, 2996, 6325, 6667, 3509, 3689, 7747, 8125, 4256, 4454, 9313, 9727, 5075, 5291

Offset: 0

Paul Curtz, Dec 21 2010

         For n =  0,  1,  2,  3,   4,   5,   6,   7, ...,
        a(n-1) = -1,  7, 25, 26,  44, 133, 187, 125, ...
  + A177049(n) =  1,  5, 14, 55,  91,  68,  95, 253, ...
         gives    0, 12, 39, 81, 135, 201, 282, 378, ...
which are increasing multiples of 3.
a(n) mod 9 = period 4: repeat 7,7,8,8.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A060819, A176672, A177049.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) )); // G. C. Greubel, Feb 26 2020

m:=50; S:=series((7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 26 2020

LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1}, {7,25,26,44,133,187,125,161,403}, 50] (* Harvey P. Dale, May 21 2015 *)

Vec( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) +O('x^50) ) \\ G. C. Greubel, Feb 26 2020

def A178370_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) ).list()
A178370_list(50) # G. C. Greubel, Feb 26 2020

a(n) = A060819(2+3*n)*(A060819(7+3*n) + A176672(n+2))/2. - corrected by G. C. Greubel, Feb 26 2020
G.f.: (7 +4*x -7*x^2 +46*x^3 -9*x^4 +8*x^5 +4*x^6 +2*x^7 -x^8)/((1-x)^3 * (1 + x^2)^3 ). - R. J. Mathar, Jan 16 2011
From G. C. Greubel, Feb 26 2020: (Start)
a(n) = (6 + i^n*(1 - i + (-1)^n*(1 + i)))*(9*n^2 + 27*n + 14)/16.
E.g.f.: ( 3*(14+36*x+9*x^2)*exp(x) + (14+36*x-9*x^2)*cos(x) + (14-36*x-9*x^2)*sin(x) )/8. (End)
Sum_{n>=0} 1/a(n) = 1 - (3 + 4*sqrt(3))*Pi/45. - Amiram Eldar, Aug 12 2022

More terms from Jinyuan Wang, Feb 26 2020

A302773 Numerators of (3*n + 2)/12.

Original entry on oeis.org

1, 5, 2, 11, 7, 17, 5, 23, 13, 29, 8, 35, 19, 41, 11, 47, 25, 53, 14, 59, 31, 65, 17, 71, 37, 77, 20, 83, 43, 89, 23, 95, 49, 101, 26, 107, 55, 113, 29, 119, 61, 125, 32, 131, 67, 137, 35, 143, 73, 149, 38, 155, 79, 161, 41, 167, 85, 173, 44, 179, 91, 185, 47, 191, 97

Offset: 0

Bruno Berselli, Apr 13 2018

Or numerators of (3*n+2)/4. - Altug Alkan, Apr 17 2018

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

Cf. A016789, A109008.
Cf. A060819: numerators of n/4, with n > 0.
Cf. A176672: numerators of (3*n + 1)/12.
First bisection is A165355; second bisection is A016969.

List([0..70], n -> NumeratorRat((3*n+2)/12));

Magma
```
[Numerator((3*n+2)/12): n in [0..70]];
```

Table[Numerator[(3 n + 2)/12], {n, 0, 70}]
LinearRecurrence[{0,0,0,2,0,0,0,-1},{1,5,2,11,7,17,5,23},80] (* Harvey P. Dale, Feb 04 2021 *)

vector(70, n, n--; numerator((3*n+2)/12))

Vec((1 + 5*x + 2*x^2 + 11*x^3 + 5*x^4 + 7*x^5 + x^6 + x^7)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Apr 16 2018

[numerator((3*n+2)/12) for n in (0..70)]

G.f.: (1 + 5*x + 2*x^2 + 11*x^3 + 5*x^4 + 7*x^5 + x^6 + x^7)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8).
a(n) = (3*n + 2)*(((-1)^n + 1)*(i^(n*(n+1)) - 5) + 16)/16, where i = sqrt(-1).
a(n) = A016789(n)/A109008(n+2).

cp's OEIS Frontend

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

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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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A100652 Denominator of 1 - Sum_{i=1..n} |Bernoulli(i)|.

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A178370 The trisection A178242(3n+2).

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A302773 Numerators of (3*n + 2)/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.