A173973 -id:A173973 - cp's OEIS frontend

A003983 Array read by antidiagonals with T(n,k) = min(n,k).

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

Offset: 1

Marc LeBrun

Also, "correlation triangle" for the constant sequence 1. - Paul Barry, Jan 16 2006
Antidiagonal sums are in A002620.
As a triangle, row sums are A002620. T(2n,n)=n+1. Diagonal sums are A001399. Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the constant sequence 1 (lower triangular matrix with all 1's). - Paul Barry, Jan 16 2006
From Franklin T. Adams-Watters, Sep 25 2011: (Start)
As a triangle, count up to ceiling(n/2) and back down again (repeating the central term when n is even).
When the first two instances of each number are removed from the sequence, the original sequence is recovered.
(End)

			Triangle version begins
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 2, 3, 2, 1;
  1, 2, 3, 3, 2, 1;
  1, 2, 3, 4, 3, 2, 1;
  1, 2, 3, 4, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 4, 3, 2, 1;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Cf. A002620, A001399, A087062, A115236, A115237, A124258, A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173973, A173982-A173986, A004197.

a003983 n k = a003983_tabl !! (n-1) !! (k-1)
a003983_tabl = map a003983_row [1..]
a003983_row n = hs ++ drop m (reverse hs)
   where hs = [1..n' + m]
         (n',m) = divMod n 2
-- Reinhard Zumkeller, Aug 14 2011

a(n) = min(floor(1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2) # Leonid Bedratyuk, Dec 13 2009

Flatten[Table[Min[n-k+1, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Feb 23 2012 *)

T(n,k) = min(n,k) \\ Charles R Greathouse IV, Feb 06 2017

from math import isqrt
def A003983(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-(a*(a-1)>>1)
    return min(x,a-x+1) # Chai Wah Wu, Jun 14 2025

Number triangle T(n, k) = Sum_{j=0..n} [j<=k][j<=n-k]. - Paul Barry, Jan 16 2006
G.f.: 1/((1-x)*(1-x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006
a(n) = min(floor( 1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2). - Leonid Bedratyuk, Dec 13 2009

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Entry revised by N. J. A. Sloane, Dec 05 2006

A173982 a(n) = numerator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 9, 153, 7641, 192789, 32757741, 525987081, 190358321841, 23076404893161, 577743530648769, 578407918658769, 556370890030917009, 160916328686946575601, 220439117509451225357769

Offset: 0

Artur Jasinski, Mar 04 2010

All numbers in this sequence are divisible by 9.

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173984.
For A173982/9 see A173983.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987.

[0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

A173982 := proc(n) add( 1/(1/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173982(n),n=0..20) ; # R. J. Mathar, Apr 22 2010

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[9/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(numerator(sum(k=0,n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of 2*(Pi^2)/3 + J - Zeta(2,(3*n+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
A173982(n)/A173984(n) = sum_{i=0..n} 1/(1/3+i)^2 = 9*sum_{i=0..n} 1/(1+3i)^2 = psi'(1/3) - psi'(n+1/3). - R. J. Mathar, Apr 22 2010
a(n) = numerator of Sum_{k=0..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.

Original entry on oeis.org

0, 1, 17, 849, 21421, 3639749, 58443009, 21150924649, 2564044988129, 64193725627641, 64267546517641, 61818987781213001, 17879592076327397289, 24493235278827913928641, 24506988360923903264741

Offset: 0

Artur Jasinski, Mar 04 2010

From Wolfdieter Lang, Nov 12 2017: (Start)
a(n+1)/A173984(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(1+3*k)^2.
The limit n -> infinity is given in A214550 as the Hurwitz Zeta function or the Polygamma function (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) = 1.121733...  (End)

			The rationals a(n)/A173984(n) begin 0/1, 1/1, 17/16, 849/784, 21421/19600, 3639749/3312400, 58443009/52998400, 21150924649/19132422400, ... - _Wolfdieter Lang_, Nov 12 2017

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173984.
For 9*A173983 see A173982.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987, A214550.

[0] cat [Numerator((&+[1/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

a := n -> numer((Zeta(0,2,1/3) - Zeta(0,2,n+1/3))/9):
seq(a(n), n=0..14); # Peter Luschny, Nov 12 2017

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]]/9, {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[1/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of (1/9)(2(Pi^2)/3 + J - Zeta(2,(3n+1)/3)) where J is the constant A173973.

a(n) = numerator of Sum_{k=0..(n-1)} 1/(3*k+1)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 12 2017

A173987 a(n) = denominator of ((Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 4, 100, 1600, 193600, 9486400, 2741569600, 2741569600, 1450290318400, 245099063809600, 206128312663873600, 3298053002621977600, 3298053002621977600, 1190597133946533913600, 2001393782164123508761600

Offset: 0

Artur Jasinski, Mar 04 2010

G. C. Greubel, Table of n, a(n) for n = 0..300

For numerators see A173985.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173986.

[1] cat [Denominator((&+[9/(3*k+2)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

a := n -> (Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9:
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017

Table[FunctionExpand[(1/9)*(4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 2)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(denominator(9*sum(k=0,n-1, 1/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = denominator of 2*(Pi^2)/3 - J - Zeta(2,(3*n+2)/3), where Zeta is the Hurwitz Zeta function and J is the constant A173973.

a(n) = denominator of Sum_{k=0..(n-1)} 9/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 1, 16, 784, 19600, 3312400, 52998400, 19132422400, 2315023110400, 57875577760000, 57875577760000, 55618430227360000, 16073726335707040000, 22004931353582937760000, 22004931353582937760000

Offset: 0

Artur Jasinski, Mar 04 2010

G. C. Greubel, Table of n, a(n) for n = 0..300

For the numerators see A173982.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173983, A173985, A173986, A173987.

[1,1] cat [Denominator((&+[9/(3*k+1)^2: k in [1..n-1]])): n in [2..20]]; // G. C. Greubel, Aug 24 2018

a := n -> Zeta(0,2,1/3) - Zeta(0,2,n+1/3):
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 1)^2, {k, 1, n - 1}], {n, 0, 30}]] (* G. C. Greubel, Aug 24 2018 *)

for(n=0,20, print1(denominator(sum(k=1,n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 24 2018

a(n) = denominator of 2*(Pi^2)/3 + J - Zeta(2,(3*n+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.

a(n) = denominator of Sum_{k=1..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 24 2018

Name simplified by Peter Luschny, Nov 14 2017

A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.

Original entry on oeis.org

0, 1, 29, 489, 60769, 3026081, 884023809, 890877733, 474015890357, 80471258049933, 67921427083803253, 1089963588226225073, 1092655876391630769, 395273284628034202009, 665644988593672027490729

Offset: 0

Artur Jasinski, Mar 04 2010

a(n+1)/A173987(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(3*k+2)^2. The limit n -> infinity is given in A294967 as the Hurwitz Zeta function or the Trigamma function (1/9)*Zeta(2, 2/3) = (1/9)*Psi(1, 2/3) = 0.3404306010 ... - Wolfdieter Lang, Nov 12 2017

			The rationals a(n)/A173987(n) begin 0/1, 1/4, 29/100, 489/1600, 60769/193600, 3026081/9486400, 884023809/2741569600, 890877733/2741569600, ... - _Wolfdieter Lang_, Nov 12 2017

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173987.
For 9*A173986 see A173985.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173985, A294967.

[0] cat [Numerator((&+[2/(3*k+2)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> (Psi(1, 2/3) - Psi(1, n+2/3))/9:
seq(numer(simplify(r(n))), n=0..14); # Peter Luschny, Nov 13 2017

Table[Numerator[FunctionExpand[(4*Pi^2/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])/9]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[2/(3*k + 2)^2, {k, 0, n - 2}], {n, 1, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=1,20, print1(numerator(sum(k=0,n-2, 2/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator(r(n)) with r(n) = (1/9)*(4*(Pi^2)/3 - Zeta(2, 1/3) - Zeta(2, (3*n+2)/3)) = (1/9)*(Zeta(2, 2/3) - Zeta(2, (3*n+2)/3)) with the Hurwitz Zeta function Zeta(2, q). This becomes the formula given in the name. - Wolfdieter Lang, Nov 13 2017
a(n) = numerator of (1/9)*(2(Pi^2)/3 - J - Zeta(2, (3n+2)/3)) where J is the constant A173973 [which becomes the preceding formula].
a(n) = numerator of Sum_{k=0..(n-2)} 2/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 13 2017

A173985 a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 9, 261, 4401, 546921, 27234729, 7956214281, 8017899597, 4266143013213, 724241322449397, 611292843754229277, 9809672294036025657, 9833902887524676921, 3557459561652307818081, 5990804897343048247416561

Offset: 0

Artur Jasinski, Mar 04 2010

All numbers in this sequence are divisible by 9.

G. C. Greubel, Table of n, a(n) for n = 0..300

For denominators see A173987.
For A173985/9 see A173986.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984.

[0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

A173985 := proc(n) add( 1/(2/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173985(n),n=0..20) ; # R. J. Mathar, Apr 22 2010

Table[FunctionExpand[4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3]], {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[9/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

for(n=0,20, print1(numerator(9*sum(k=0,n-1, 1/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of 2*(Pi^2)/3 - J - Zeta(2, (3*n+2)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
a(n)/A173987(n) = sum_{i=0..n-1} 1/(i+2/3)^2 = psi'(2/3)-psi'(2/3+n). - R. J. Mathar, Apr 22 2010
a(n) = numerator of Sum_{k=0..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 23 2018

Name simplified by Peter Luschny, Nov 14 2017

A214552 Decimal expansion of the Dirichlet L-series of the non-principal character mod 6 evaluated at s=2.

Original entry on oeis.org

9, 7, 6, 6, 2, 8, 0, 1, 6, 1, 2, 0, 6, 0, 7, 8, 7, 1, 0, 8, 3, 9, 8, 4, 2, 8, 7, 0, 3, 0, 1, 1, 5, 4, 4, 5, 4, 5, 6, 4, 1, 7, 9, 2, 0, 6, 8, 1, 6, 0, 6, 7, 7, 5, 2, 7, 7, 6, 2, 5, 0, 7, 8, 7, 0, 8, 6, 0, 8, 7, 3, 0, 8, 1, 4, 5, 2, 2, 7, 7, 2, 6, 1, 6, 0, 8, 6, 9, 6, 3, 5, 4, 0, 2, 6, 2, 3, 2, 6, 2, 7, 6, 3, 0, 2

Offset: 0

R. J. Mathar, Jul 20 2012

The non-principal character is A134667. The constant is sum_{n>=1} A134667(n)/n^s with s=2.

			0.97662801612060787108398...= 1/1^2 -1/5^2 +1/7^2 -1/11^2 + 1/13^2 -1/17^2 +-...

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:10008.2547 [math.NT], 2010-2015, Table in section 2.2, value at m=6, r=2, s=2.

Cf. A086724, A086722, A100044.

Maple
```
evalf( (Psi(1,1/6)-Psi(1,5/6))/36) ;
```

RealDigits[ (PolyGamma[1, 1/6] - PolyGamma[1, 5/6])/36, 10, 105] // First  (* Jean-François Alcover, Feb 11 2013, after R. J. Mathar *)

Equals 2/3*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function. - Jean-François Alcover, Dec 16 2014, after R. J. Mathar.

Equals A173973 / 3.6 . - R. J. Mathar, Jun 02 2016

More terms from Jean-François Alcover, Feb 11 2013

cp's OEIS Frontend

A003983 Array read by antidiagonals with T(n,k) = min(n,k).

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A173982 a(n) = numerator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)), where Zeta is the Hurwitz Zeta function.

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A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.

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A173987 a(n) = denominator of ((Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9), where Zeta is the Hurwitz Zeta function.

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A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

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A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.

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