A161738 -id:A161738 - cp's OEIS frontend

A093178 If n is even then 1, otherwise n.

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

Offset: 0

Michael Somos, Mar 27 2004

Continued fraction expansion for tan(1).
1 followed by run lengths of A062557 = 2n-1 1's followed by a 2. - Jeremy Gardiner, Aug 12 2012
Greatest common divisor of n and (n+1) mod 2. - Bruno Berselli, Mar 07 2017

			1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...))))
G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 5*x^5 + x^6 + 7*x^7 + x^8 + 9*x^9 + x^10 + ...

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Simon Plouffe, A Search for a mathematical expression for mass ratios using a large database. page 3.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Equals |A009001(n)|.

Cf. A133080, A049471 (decimal expansion), A009001, A161738, A062557, A124625.

A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* Harvey P. Dale, Nov 23 2011 *)

PARI
```
{a(n) = if( n%2, n, 1)};
```

G.f.: (1+x-x^2+x^3)/(1-x^2)^2.
a(n) = (-1)^n * a(-n) for all n in Z.
a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - Ralf Stephan, Dec 02 2004
a(n) = n^n mod (n+1) for n > 0. - Amarnath Murthy, Apr 18 2004
Satisfies a(0) = 1, a(n+1) = a(n) + n if a(n) < n else a(n+1) = a(n)/n. - Amarnath Murthy, Oct 29 2002
a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - Paul Barry, Apr 09 2003
a(n) = binomial(n, 2*floor(n/2)). - Paul Barry, Dec 28 2006
Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
a(n) = denom(b(n+2)/b(n+1)) with b(n) = product((2*n-3-2*k), k=0..floor(n/2-1)). - Johannes W. Meijer, Jun 18 2009
a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - Wesley Ivan Hurt, Oct 19 2013
a(n) = n^(n mod 2). - Wesley Ivan Hurt, Apr 16 2014

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

A161737 Numerators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 2, 16, 128, 2048, 32768, 262144, 2097152, 67108864, 2147483648, 17179869184, 137438953472, 2199023255552, 35184372088832, 281474976710656, 2251799813685248, 144115188075855872, 9223372036854775808, 73786976294838206464, 590295810358705651712, 9444732965739290427392

Offset: 1

Johannes W. Meijer, Jun 18 2009

For the definition of the BG2 matrix coefficients see A161736.

			sb(1) = 1; sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 1..830

Cf. A161736 and A161738, A050605, A007814 and A090739.

[Numerator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; x(1):=0: x(2):=1: for n from 2 to nmax-1 do x(n+1) := A050605(n-2) + x(n) + 3 od: for n from 1 to nmax do a(n) := 2^x(n) od: seq(a(n), n=1..nmax); # End program 1
nmax1 := 20; for n from 0 to nmax1 do y(2*n+1) := A090739(n); y(2*n) := A090739(n) od: z(1) := 0: z(2) := 1: for n from 3 to nmax1 do z(n) := z(n-1) + y(n-1) od: for n from 1 to nmax1 do a(n) := 2^z(n) od: seq(a(n), n=1..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012 and by Peter Luschny, Feb 13 2025
r := n -> Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> denom(simplify(r(n))):
seq(a(n), n = 1..20);  # Peter Luschny, Feb 12 2025

sb[1] = 1; sb[2] = 2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2;
Table[sb[n] // Numerator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

vector(20, n, n++; numerator((2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2))) \\ G. C. Greubel, Sep 26 2018

a(n) = numer(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161736(n) = denom(sb(n)).

a(n) = denominator(Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 12 2025

Offset set to 1 and a(1) = 1 prepended by Peter Luschny, Feb 13 2025

cp's OEIS Frontend

A093178 If n is even then 1, otherwise n.

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A161736 Denominators of the column sums of the BG2 matrix.

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Magma

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A161737 Numerators of the column sums of the BG2 matrix.

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Magma

Maple

Mathematica

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Formula

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