A130823 -id:A130823 - cp's OEIS frontend

A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1

Offset: 1

Omar E. Pol, Jul 02 2012

Column k lists the odd numbers repeated k times starting in row k^2.
1 together with the first differences of the row sums give the divisor function A000005.
T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;

Omar E. Pol, Diagram of divisors, figure 1, figure 2.

Row sums give A006218, n >= 1.
Columns (1-5): A005408, A109613, A130823, A129756, A130497.
Cf. A000005, A027750, A010766, A147861, A163100, A212119.

T(n,k) = Sum_{j=1..n} A212119(j,k).

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A166107 A sequence related to the Madhava-Gregory-Leibniz formula for Pi.

Original entry on oeis.org

2, -10, 46, -334, 982, -10942, 140986, -425730, 7201374, -137366646, 410787198, -9473047614, 236302407090, -710245778490, 20563663645710, -638377099140510, 1912749274005030, -67020067316087550, 2477305680740159850

Offset: 0

Johannes W. Meijer, Oct 06 2009, Feb 26 2013, Mar 02 2013

The EG1 matrix is defined in A162005. The first column of this matrix leads to the function PLS(z) = sum(2*eta(2*m-1)*z^(2*m-2), m=1..infinity) = 2*log(2) - Psi(z) - Psi(-z) + Psi(z/2) + Psi(-z/2). The values of this function for z=n+1/2 are related to Pi in a curious way.

Gauss's digamma theorem leads to PLS(z=n+1/2) = (-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1). Now we define PLS(z=n+1/2) = a(n)/p(n) with a(n) the sequence given above and for p(n) we choose the esthetically nice p(n) = (2*n-1)!!/(floor((n-2)/3)*2+1)!!, n=>0. For even values of n the limit(a(2*n)/p(2*n), n=infinity) = Pi and for odd values of n the limit(a(2*n+1)/p(2*n+1), n=infinity) = - Pi. We observe that the a(n)/p(n) formulas resemble the partial sums of the Madhava-Gregory-Leibniz series for Pi = 4*(1-1/3+1/5-1/7+ ...), see the examples. The 'extra term' that appears in the a(n)/p(n) formulas, i.e., 2/(2*n+1), speeds up the convergence of abs(a(n)/p(n)) significantly. The first appearance of a digit in the decimal expansion of Pi occurs here for n: 1, 3, 9, 30, 74, 261, 876, 3056, .., cf. A126809. [Comment modified by the author, Oct 09 2009]

			The first few values of a(n)/p(n) are: a(0)/p(0) = 2/1; a(1)/p(1) = - 4*(1) + 2/3 = -10/3; a(2)/p(2) = 4*(1-1/3) + 2/5 = 46/15; a(3)/p(3) = - 4*(1-1/3+1/5) + 2/7 = - 334/105; a(4)/p(4)= 4*(1-1/3+1/5-1/7) + 2/9 = 982/315; a(5)/p(5) = - 4*(1-1/3+1/5-1/7+1/9) + 2/11 = -10942/3465; a(6)/p(6) = 4*(1-1/3+1/5-1/7+1/9-1/11) + 2/13 = 140986/45045; a(7)/p(7) = - 4*(1-1/3+1/5-1/7+1/9-1/11+1/13) + 2/15 = - 425730/135135.

Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.
Johannes W. Meijer, A modified Madhava-Gregory-Leibniz formula for Pi, Mar 02 2013.
Pacific Institute for the Mathematical Sciences, Pi in the Sky magazine, 2000-2013.
Wislawa Szymborska, The admirable number Pi, Miracle Fair, 2002.
Eric. W. Weisstein, Gauss's Digamma Theorem., from Wolfram MathWorld.
Wikipedia, Madhava of Sangamagrama.

Cf. A162005, A001147, A130823, A025547, A220747, A000796, A157142, A126809, A133766, A133767, A154633.

A166107 := n -> A220747 (n)*((-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1)): A130823 := n -> floor((n-1)/3)*2+1: A220747 := n -> doublefactorial(2*n+1) / doublefactorial(A130823(n)): seq(A166107(n), n=0..20);

a(n) = p(n)*(-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1) with
p(n) = doublefactorial(2*n+1)/doublefactorial(floor((n-1)/3)*2+1) = A220747(n)
PLS(z) = 2*log(2) - Psi(z) - Psi(-z) + Psi(z/2) + Psi(-z/2)
PLS(z=n+1/2) = a(n)/p(n) = (-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1)
PLS(z=2*n+5/2) - PLS(z=2*n+1/2) = 2/(4*n+5) - 4/(4*n+3) + 2/(4*n+1) which leads to:
Pi = 2 + 16 * sum(1/((4*n+5)*(4*n+3)*(4*n+1)), n=0 .. infinity).
PLS (z=2*n +7/2) - PLS(z=2*n+3/2) = 2/(4*n+7) - 4/(4*n+5) + 2/(4*n+3) which leads to:
Pi = 10/3 - 16*sum(1/((4*n+7)*(4*n+5)*(4*n+3)), n=0 .. infinity).
The combination of these two formulas leads to:
Pi = 8/3 + 48* sum(1/((4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)), n=0 .. infinity).

A215495 a(4n) = a(4n+2) = a(2n+1) = 2n + 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 7, 13, 7, 15, 9, 17, 9, 19, 11, 21, 11, 23, 13, 25, 13, 27, 15, 29, 15, 31, 17, 33, 17, 35, 19, 37, 19, 39, 21, 41, 21, 43, 23, 45, 23, 47, 25, 49, 25, 51, 27, 53, 27, 55, 29, 57, 29, 59, 31, 61, 31, 63, 33, 65, 33, 67, 35, 69, 35, 71, 37, 73, 37, 75, 39, 77, 39, 79, 41, 81, 41, 83, 43, 85, 43

Offset: 0

Paul Curtz, Aug 13 2012

A214282(n) and -A214283(n) are companions. Separately or together, they have many links with the Catalan's numbers A000108(n). Examples:
A214282(n+1) - 2*A214282(n) = -1, -1,  1,  0, -2, -5,  5,   0, -14, -42,  42,    0, -132, ....
2*A214283(n) - A214283(n+1) =  1,  0, -1, -2,  2,  0, -5, -14,  14,   0, -42, -132, 132, ....
A214282(n) + A214283(n) = 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42,... (A126120).
The companion to a(n) is b(n) = -A214283(n)/(1,1,1,1,2,2,5,5,...) = 0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, ....
a(n) - b(n) = A056594(n).
Discovered as a(n) = A214282(n+1)/A000108([n/2]). See abs(A129996(n-2)).

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

Cf. A130823, A090192.

I:=[1,1,1,3,3,5]; [n le 6 select I[n] else Self(n-2) +Self(n-4) -Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 23 2018

a[n_?EvenQ] := n/2 + Boole[Mod[n, 4] == 0]; a[n_?OddQ] := n; Table[a[n], {n, 0, 86}] (* Jean-François Alcover, Aug 14 2012 *)
LinearRecurrence[{0,1,0,1,0,-1}, {1,1,1,3,3,5}, 50] (* G. C. Greubel, Apr 23 2018 *)

x='x+O('x^30); Vec(( 1+x+2*x^3+x^4+x^5 )/( (x^2+1)*(x-1)^2*(1+x)^2 )) \\ G. C. Greubel, Apr 23 2018

a(n+3) = (A185048(n+3)=2,2,4,2,... ) + 1.
a(n+2) - a(n) = 0, 2, 2, 2. (Period 4).
a(n) = 2*a(n-4) - a(n-8).
a(2*n) = A109613(n).
a(n+1) - a(n) = 2* (-1)^n * A059169(n).
G.f. : ( 1+x+2*x^3+x^4+x^5 ) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - Jean-François Alcover, Aug 14 2012

A226023 A142705 (numerators of 1/4-1/(4n^2)) sorted to natural order.

Original entry on oeis.org

0, 2, 3, 6, 12, 15, 20, 30, 35, 42, 56, 63, 72, 90, 99, 110, 132, 143, 156, 182, 195, 210, 240, 255, 272, 306, 323, 342, 380, 399, 420, 462, 483, 506, 552, 575, 600, 650, 675, 702, 756, 783, 812, 870, 899

Offset: 0

Paul Curtz, May 23 2013

A198442(n) without indices 4*n+2.
a(n)/A130823(n+1) = 0, 2,3,2, 4,5,4, 6,7,6, 8,9,8, ... (equal to A133310+1, after 0; see also A008611).
-1,   0,   2,   3, is divisible by  1 (for a(-1)=-1),
3,    6,  12,  15,                  3,
15,  20,  30,  35                   5,
35,  42,  56,  63                   7,
63,  72,  90,  99                   9,
99, 110, 132, 143,                 11, etc.
First column:  A000466(n),
second column: A002943(n),
third column:  A002939(n+1),
fourth column: A000466(n+1).
a(n) is also the numerator of 1/4-1/(4*n+2)^2: 0/1, 2/9, 3/16, 6/25, 12/49, 15/64, 20/81, 30/121, 35/144, 42/169, 56/225,...
The n-th denominator is equal to 4*a(n) + A146325(n+2).
Note that the differences of a(n-1): 1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22,... (from A043547 by pairs and 2*n+1) has the same recurrence.
(Of course every sequence which obeys a linear recurrence with constant coefficients has first differences that obey the same linear recurrence. - R. J. Mathar, Jun 14 2013)

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Trisections: A002939, A000466, A002943.

A226023 := proc(n)
    option remember;
    if n <=6 then
        op(n+1,[0,2,3,6,12,15,20]) ;
    else
        procname(n-1)+2*procname(n-3)-2*procname(n-4)-procname(n-6)+procname(n-7) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

A226023[n_]:=Floor[(2n+1)/3]Floor[(2n+5)/3];
Array[A226023,100,0] (* Paolo Xausa, Dec 05 2023 *)

a(n) = floor( (2*n + 1)/3 ) * floor( (2*n + 5)/3 ) = A004396(n) * A004396(n+2).
Recurrences: a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7).
a(n+15)  - a(n) = 10*A042968(n+8).
a(n+1) - a(n-2) = 2*A042968(n) with a(-2)=0, a(-1)=-1.
G.f.: x*(2+x+3*x^2+2*x^3+x^4-x^5)/((1-x)^3 * (1+x+x^2)^2). [Ralf Stephan, May 24 2013]

A133310 a(3n) = 2n+1, a(3n+1) = 2n+2, a(3n+2) = 2n+1.

Original entry on oeis.org

1, 2, 1, 3, 4, 3, 5, 6, 5, 7, 8, 7, 9, 10, 9, 11, 12, 11, 13, 14, 13, 15, 16, 15, 17, 18, 17, 19, 20, 19, 21, 22, 21, 23, 24, 23, 25, 26, 25, 27, 28, 27, 29, 30, 29, 31, 32, 31, 33, 34, 33, 35, 36, 35, 37, 38, 37, 39, 40, 39, 41, 42, 41, 43, 44, 43, 45, 46, 45, 47, 48, 47, 49, 50

Offset: 0

Paul Curtz, Oct 18 2007

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

Cf. A130823 (1, 1, 1, 3, 3, 3).

Q:=Rationals(); R:=PowerSeriesRing(Q, 70); Coefficients(R!((1+x-x^2+x^3)/((1+x+x^2)*(x-1)^2))) // G. C. Greubel, Feb 10 2018

CoefficientList[Series[(1+x-x^2+x^3)/((1+x+x^2)*(x-1)^2), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 2, 1, 3}, 74] (* Robert G. Wilson v, Feb 10 2018 *)

my(x='x+O('x^70)); Vec((1+x-x^2+x^3)/((1+x+x^2)*(x-1)^2)) \\ G. C. Greubel, Feb 10 2018

G.f.: ( 1+x-x^2+x^3 ) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, May 23 2014

a(n) = (2/9)*(3*n+3+3*cos(2*(n-1)*Pi/3)-2*sqrt(3)*sin(2*(n-1)*Pi/3)). - Wesley Ivan Hurt, Sep 30 2017

A220747 a(n) = (2n+1)!! / ((floor((n-1)/3)2+1))!!

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 135135, 2297295, 43648605, 130945815, 3011753745, 75293843625, 225881530875, 6550564395375, 203067496256625, 609202488769875, 21322087106945625, 788917222956988125

Offset: 0

Johannes W. Meijer, Feb 26 2013

The a(n) appear in the analysis of a sequence that is related to the Madhava-Gregory-Leibniz formula for Pi, see A166107.

Cf. A000796, A001147, A130823, A166107.

A220747 := n -> doublefactorial(2*n+1)/doublefactorial(A130823(n)): A130823 := n -> floor((n-1)/3)*2+1: seq(A220747(n), n=0..20);

Table[(2*n + 1)!!/((Floor[(n - 1)/3]*2 + 1))!!, {n, 0, 20}] (* T. D. Noe, Feb 26 2013 *)
CoefficientList[Series[HypergeometricPFQ[{5/6, 7/6}, {1/3, 2/3}, 4 x^3] + 3/2 x (2 HypergeometricPFQ[{5/6, 7/6}, {2/3, 4/3}, 4 x^3] + 5x HypergeometricPFQ[{7/6, 11/6}, {4/3, 5/3}, 4 x^3]), {x, 0, 20}], x]*Range[0, 20]! (* Benedict W. J. Irwin, Oct 19 2016 *)

Limit_{n -> infinity} A166107(2*n)/a(2*n) = Pi.
Limit_{n -> infinity} A166107(2*n+1)/a(2*n+1) = -Pi.
E.g.f.: 2F2(5/6,7/6; 1/3,2/3; 4*x^3) + 3*x*(2F2(5/6,7/6; 2/3,4/3; 4*x^3) + 5*x*2F2(7/6,11/6; 4/3,5/3; 4*x^3)/2). - Benedict W. J. Irwin, Oct 19 2016

A329583 Numerators of 1 + n^2/4 + period 3: repeat [-1, 1, 1].

Original entry on oeis.org

0, 6, 3, 12, 6, 30, 9, 54, 18, 84, 27, 126, 36, 174, 51, 228, 66, 294, 81, 366, 102, 444, 123, 534, 144, 630, 171, 732, 198, 846, 225, 966, 258, 1092, 291, 1230, 324, 1374, 363, 1524, 402, 1686, 441, 1854, 486, 2028, 531, 2214, 576, 2406, 627

Offset: 0

Paul Curtz, Nov 17 2019

First bisection is 3*A008810.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2,3,0,-3,-2,1,1).

Cf. A008810, A042965, A047395, A130823, A131561, A261327.

MapIndexed[#1 - 2 Boole[Mod[First@ #2, 3] == 1] + 1 &, CoefficientList[Series[(1 + 5 x - x^2 - 2 x^3 + 2 x^4 + 5 x^5)/(1 - x^2)^3, {x, 0, 44}], x]] (* Michael De Vlieger, Nov 18 2019 *)

concat(0, Vec(3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Nov 24 2019

a(n) = A261327(n) + A131561(n+2) = (n^2 + 4)*(5 - 3*(-1)^n)/8 + (-1)^((n+1) mod 3).
From Colin Barker, Nov 24 2019: (Start)
G.f.: 3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)).
a(n) = -a(n-1) + 2*a(n-2) + 3*a(n-3) - 3*a(n-5) - 2*a(n-6) + a(n-7) + a(n-8) for n>8. (End)

Incorrect 129 replaced with 123 by Colin Barker, Nov 24 2019

A226122 Expansion of (1+2x+x^2+x^3+2x^4+x^5)/(1-2*x^3+x^6).

Original entry on oeis.org

1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22, 11, 13, 26, 13, 15, 30, 15, 17, 34, 17, 19, 38, 19, 21, 42, 21, 23, 46, 23, 25, 50, 25, 27, 54, 27, 29, 58, 29, 31, 62, 31, 33, 66, 33, 35, 70, 35, 37, 74, 37, 39, 78, 39

Offset: 0

Paul Curtz, May 27 2013

A226023 (starting from A226023(-2)=0) and successive differences:
0,    -1,   0,    2,     3,   6,   12,   15,   20,   30,...
-1,    1,   2,    1,     3,   6,    3,    5,   10,    5,...  = a(n-1)
2,     1,  -1,    2,     3,  -3,    2,    5,   -5,    2,...
-1,   -2,   3,    1,    -6,   5,    3,  -10,    7,    5,...
-1,    5,  -2,   -7,    11,  -2,  -13,   17,   -2,  -19,...
6,    -7,  -5,   18,   -13, -11,   30,  -19,  -17,   42,...
-13,   2,  23,  -31,    2,   41,  -49,    2,   59,   67,...
15,   21, -54,   33,   39,  -90,   51,   57, -126,   69,... multiples of 3
6,   -75,  87,    6, -129,  141,    6, -183,  195,    6,... multiples of 3
-81, 162, -81, -135,  270, -135, -189,  378, -189, -243,... multiples of 27
The last line is -27*a(n+3)*A131561(n+1).
The recurrences in the Formula field hold for the array.

			Given A130823 = 1,1,1,3,3,3,5,5,5,7,7,7,... and A131534 = 1,2,1,1,2,1,1,2,1,1,2,1,..., then a(0)=1*1=1, a(1)=1*2=2, a(2)=1*1=1, a(3)=3*1=3, a(4)=3*2=6, etc.
Given A226023(n) from A226023(-1)=-1, then a(0)=0-(-1)=1, a(1)=2-0=2, a(2)=3-2=1, a(3)=6-3=3, a(4)=12-6=6, etc.

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

Cf. A005408, A016754, A130823, A131534, A226023.

repeat=20; Table[{1, 2, 1}, {repeat}]*(2*Range[repeat]-1) // Flatten
(* or *) Table[Floor[(2*n+1)/3]*Floor[(2*n+5)/3], {n, -1, 59}] // Differences (* Jean-François Alcover, May 29 2013 *)

a(n) = A130823(n-1) * A131534(n).
a(n) = A226023(n) - A226023(n-1) with A226023(-1)=-1.
a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7). [Ralf Stephan]
From Bruno Berselli, May 29 2013: (Start)
G.f.: (1+x)^3*(1-x+x^2)/((1-x)^2*(1+x+x^2)^2).
a(n) = 2*a(n-3)-a(n-6).
a(3n)*a(3n-1)-a(3n-2) = A016754(n-1), n>0. (End)

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A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

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A166107 A sequence related to the Madhava-Gregory-Leibniz formula for Pi.

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A215495 a(4*n) = a(4*n+2) = a(2*n+1) = 2*n + 1.

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A226023 A142705 (numerators of 1/4-1/(4n^2)) sorted to natural order.

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A133310 a(3n) = 2n+1, a(3n+1) = 2n+2, a(3n+2) = 2n+1.

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A220747 a(n) = (2*n+1)!! / ((floor((n-1)/3)*2+1))!!

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A329583 Numerators of 1 + n^2/4 + period 3: repeat [-1, 1, 1].

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A226122 Expansion of (1+2*x+x^2+x^3+2*x^4+x^5)/(1-2*x^3+x^6).

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A215495 a(4n) = a(4n+2) = a(2n+1) = 2n + 1.

A220747 a(n) = (2n+1)!! / ((floor((n-1)/3)2+1))!!

A226122 Expansion of (1+2x+x^2+x^3+2x^4+x^5)/(1-2*x^3+x^6).