Fabian Rothelius - cp's OEIS frontend

A059920 If m/n = q + r/n (r < n, n,m >=1), then array a(m,n) = qr (meaning q followed by r). Read by antidiagonals.

10, 1, 20, 1, 10, 30, 1, 2, 11, 40, 1, 2, 10, 20, 50, 1, 2, 3, 11, 21, 60, 1, 2, 3, 10, 12, 30, 70, 1, 2, 3, 4, 11, 20, 31, 80, 1, 2, 3, 4, 10, 12, 21, 40, 90, 1, 2, 3, 4, 5, 11, 13, 22, 41, 100, 1, 2, 3, 4, 5, 10, 12, 20, 30, 50, 110, 1, 2, 3, 4, 5, 6, 11, 13, 21, 31, 51, 120, 1, 2, 3

Offset: 1

Fabian Rothelius, Feb 09 2001

			a(7,3)=21 because 7/3=2+1/3; a(273,24)=119 because 273/24=11+9/24.
Array begins
10 20 30 40 50...
1 10 11 20 21 ...
1 2 10 11 ...

A059922 Each term in the table is the product of the two terms above it + 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 41, 41, 5, 1, 1, 6, 206, 1682, 206, 6, 1, 1, 7, 1237, 346493, 346493, 1237, 7, 1, 1, 8, 8660, 428611842, 120057399050, 428611842, 8660, 8, 1, 1, 9, 69281, 3711778551721, 51458022952549550101, 51458022952549550101, 3711778551721, 69281, 9, 1

Offset: 0

Fabian Rothelius, Feb 09 2001

Row sums are A059731.

			Triangle begins:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,10,4,1; ...

S. Kak, The Golden Mean and the Physics of Aesthetics

Cf. A007318, A059730 - A059733.

a059922 n k = a059922_tabl !! n !! k
a059922_flattened = concat a059922_tabl
a059922_tabl = iterate (\rs ->
   zipWith (+) (0 : reverse (0 : replicate (length rs - 1) 1))
               $ zipWith (*) ([1] ++ rs) (rs ++ [1])) [1]
a059730 n = a059922_tabl !! n !! (n-3)
a059731 n = sum (a059922_tabl !! n)
a059732 n = a059922_tabl !! (2*n) !! n
a059733 n = a059922_tabl !! n !! n `div` 2
-- Reinhard Zumkeller, Jun 22 2011

aaa := proc(m,n) option remember; if n>m or n<0 then 0; elif m=0 and n=0 then 1; else aaa(m-1,n-1)*aaa(m-1,n)+1; fi; end;

a[0, 0] = 1; a[m_, n_] /; (n > m || n < 0) = 0; a[m_, n_] := a[m, n] = a[m-1, n-1]*a[m-1, n] + 1; Table[a[m, n], {m, 0, 9}, {n, 0, m}] // Flatten (* Jean-François Alcover, Sep 10 2013 *)

a(m, n) = a(m-1, n-1)*a(m-1, n)+1, a(0, 0) = 1, a(m, n) = 0 iff n>m or n<0.

More terms from N. J. A. Sloane and Larry Reeves, Feb 09 2001.

Corrected by Jonathan Wellons (wellons(AT)gmail.com), May 24 2008

A059674 Square array a(m,n) = binomial(max(m,n), min(m,n)) (m>=0, n>=0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 4, 3, 3, 4, 1, 1, 5, 6, 1, 6, 5, 1, 1, 6, 10, 4, 4, 10, 6, 1, 1, 7, 15, 10, 1, 10, 15, 7, 1, 1, 8, 21, 20, 5, 5, 20, 21, 8, 1, 1, 9, 28, 35, 15, 1, 15, 35, 28, 9, 1, 1, 10, 36, 56, 35, 6, 6, 35, 56, 36, 10, 1, 1, 11, 45, 84, 70, 21, 1, 21, 70, 84

Offset: 0

Fabian Rothelius, Feb 05 2001

			a(2,4) = binomial(max(2,4), min(2,4)) = binomial(4,2) = 6.
Square begins:
1 1 1 1 1 1 ...
1 1 2 3 4 5 ...
1 2 1 3 6 10...
1 3 3 1 4 10 ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A007318.

a[m_, n_] := If[m >= n, Binomial[m, n], Binomial[n, m]]; Table[a[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* Jean-François Alcover, Oct 10 2012 *)

Square array equals A007318 + transpose(A007318) - I, where I denotes the infinite identity matrix. - Peter Bala, Aug 11 2015

More terms from Larry Reeves (larryr(AT)acm.org), Feb 06 2001

A059232 a(1)= 1, a(n) = (a(n-1)^a(n-1)) + n.

Original entry on oeis.org

1, 3, 30, 205891132094649000000000000000000000000000004

Offset: 1

Fabian Rothelius, Jan 20 2001

nonn

The next term is too large to include.

			a(2) = 1^1 + 2 = 3.
a(3) = 3^3 + 3 = 27 + 3 = 30.

RecurrenceTable[{a[1]==1,a[n]==a[n-1]^a[n-1]+n},a,{n,4}] (* Harvey P. Dale, Dec 27 2012 *)

{ for (n = 1, 4, a=if (n==1, 1, a^a + n); write("b059232.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 25 2009

A059573 Polyotessamino numbers T(n,k) (1<=k<=n): take all the polyominoes with n cells (including those with holes but excluding rotations and reflections); fill them as completely as possible with rectangular 1 X k tiles; T(n,k) is number of ways of doing this.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 8, 4, 1, 12, 34, 18, 4, 1, 35, 85, 61, 22, 5, 1

Offset: 1

Fabian Rothelius, Jan 22 2001

Warning: This appears to be an erroneous version of A308437. - R. J. Mathar, May 27 2019

			1;
1,1;
2,4,1;
5,8,4,1;
12,34,18,4,1;
35,85,61,22,5,1; ...

Fabian Rothelius, The polyotessamino numbers [broken link?]

Cf. A000105.

T(n,1) = A000105(n).

A059527 Decimal expansion of imaginary part of solution to z = log z.

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Jan 21 2001

Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
Wolfram Research, FixedPoint

Real part is A059526.

Cf. A030178.

RealDigits[ Im[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Im[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)

z=I;for(k=1,16000,z=log(z));imag(z)  \\ Using realprecision \p 2010. - Stanislav Sykora, Jun 07 2015

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); imag(z) \\ Jeremy Tan, Sep 23 2017

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002

A059526 Decimal expansion of real part of solution to z = log z.

Original entry on oeis.org

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

Offset: 0

Fabian Rothelius, Jan 21 2001

Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.
A complex number w with a negative imaginary part will converge to the conjugate of z since log(conjugate(w)) = conjugate(log(w)). - Gerald McGarvey, Mar 02 2009
This z and its conjugate are the only two complex solutions of z=log(z) on the principal branch of log(z), and of exp(z)=z for |arg(z)| <= Pi. They are also the only nontrivial (z!=0) principal branch solutions of z=W(z^2), W being the Lambert W-function. Though the two values are iterative attractors of the mapping z->log(z), the convergence is rather slow; the precision improves by slightly more than one binary bit every 2.25 iterations (about 7500 iterations are needed to make stable the first 1000 decimal digits). - Stanislav Sykora, Jun 07 2015

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
Wolfram Research, FixedPoint

Imaginary part is A059527.
Cf. A030178.
Cf: A277681 (another fixed point of exp(z)).

RealDigits[ Re[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Re[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)
RealDigits[Re[x/.FindRoot[x-Log[x]==0,{x,.5,1},WorkingPrecision->200]],10,120][[1]] (* Harvey P. Dale, Aug 07 2022 *)

z=I;for(k=1,16000,z=log(z));real(z) \\ Stanislav Sykora, Jun 07 2015 \\ Using realprecision \p 2010

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); real(z) \\ Jeremy Tan, Sep 23 2017

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002

A059124 Number of letters in n (in Swedish).

Original entry on oeis.org

4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 4, 7, 7, 6, 6, 7, 5, 6, 5, 8, 8, 8, 9, 8, 8, 8, 9, 8, 7, 10, 10, 10, 11, 10, 10, 10, 11, 10, 6, 9, 9, 9, 10, 9, 9, 9, 10, 9, 6, 9, 9, 9, 10, 9, 9, 9, 10, 9, 6, 9, 9, 9, 10, 9, 9, 9, 10, 9, 7, 10, 10, 10, 11, 10, 10, 10, 11, 10, 5, 8, 8, 8, 9, 8, 8, 8, 9, 8, 6

Offset: 0

Fabian Rothelius, Feb 14 2001

			noll, ett, två, tre, fyra, fem, sex, sju, åtta, nio, tio, elva, ...

Index entries for sequences related to number of letters in n

A060473 a(n) = numerator of phi(n)/(n+1), where phi(n) is Euler's phi, A000010.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 4, 6, 2, 1, 8, 8, 6, 9, 8, 6, 10, 11, 8, 10, 4, 9, 12, 14, 8, 15, 16, 10, 16, 2, 12, 18, 6, 3, 16, 20, 12, 21, 4, 12, 22, 23, 16, 21, 20, 8, 24, 26, 18, 5, 8, 18, 28, 29, 16, 30, 10, 9, 32, 8, 20, 33, 32, 22, 24, 35, 24, 36, 12, 10, 36, 10, 24, 39, 32, 27, 40, 41

Offset: 1

Fabian Rothelius, Mar 16 2001

a(A203966(n)) = 1. - Robert G. Wilson v, Jul 05 2014

			a(7) = 3 because phi(7)/(7+1) = 6/8 = 3/4.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000010, A060474.

with(numtheory,phi): seq(numer(phi(n)/(n+1)), n=1..50);

Numerator/@Table[EulerPhi[n]/(n+1),{n,90}] (* Harvey P. Dale, May 11 2011 *)

{ for (n=1, 1000, write("b060473.txt", n, " ", numerator(eulerphi(n)/(n + 1))); ) } \\ Harry J. Smith, Jul 05 2009

More terms from Asher Auel, Mar 16 2001

A060474 a(n) = denominator of phi(n)/(n+1), where phi(n) is Euler's phi, A000010.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 5, 2, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 9, 14, 29, 15, 31, 16, 33, 17, 35, 3, 37, 19, 13, 5, 41, 21, 43, 22, 9, 23, 47, 24, 49, 25, 51, 13, 53, 27, 55, 7, 19, 29, 59, 30, 61, 31, 21, 16, 65, 11, 67, 34, 69, 35, 71, 36, 73, 37, 25, 19, 77, 13, 79, 40

Offset: 1

Fabian Rothelius, Mar 16 2001

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000010, A060473.

with(numtheory,phi): seq(denom(phi(n)/(n+1)), n=1..50);

Denominator[Table[EulerPhi[n]/(n+1),{n,80}]] (* Harvey P. Dale, Apr 13 2012 *)

{ for (n=1, 1000, write("b060474.txt", n, " ", denominator(eulerphi(n)/(n + 1))); ) } \\ Harry J. Smith, Jul 13 2009

from sympy import totient, gcd
def A060474(n): return (n+1)//gcd(n+1,totient(n)) # Chai Wah Wu, Apr 02 2021

More terms from Asher Auel, Mar 16 2001

A Maple program that should have been a PARI program removed by Harry J. Smith, Jul 13 2009

cp's OEIS Frontend

User: Fabian Rothelius

A059920 If m/n = q + r/n (r < n, n,m >=1), then array a(m,n) = qr (meaning q followed by r). Read by antidiagonals.

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A059922 Each term in the table is the product of the two terms above it + 1.

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A059674 Square array a(m,n) = binomial(max(m,n), min(m,n)) (m>=0, n>=0) read by antidiagonals.

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A059232 a(1)= 1, a(n) = (a(n-1)^a(n-1)) + n.

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Mathematica

PARI

A059573 Polyotessamino numbers T(n,k) (1<=k<=n): take all the polyominoes with n cells (including those with holes but excluding rotations and reflections); fill them as completely as possible with rectangular 1 X k tiles; T(n,k) is number of ways of doing this.

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Keywords

Comments

Examples

Links

Crossrefs

Formula

A059527 Decimal expansion of imaginary part of solution to z = log z.

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Examples

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Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A059526 Decimal expansion of real part of solution to z = log z.

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Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A059124 Number of letters in n (in Swedish).

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