A085659 -id:A085659 - cp's OEIS frontend

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

1, 2, 8, 61, 819, 17417, 506609, 15582436

Offset: 1

N. J. A. Sloane, Jul 12 2003

nonn

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).

This gives the number of different values of A + A' where A runs through the matrices counted in A085656. - Max Alekseyev, Dec 13 2005

			The singular matrix
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2
is one of the three 4 X 4 matrices which are not positive definite.

Index entries for sequences related to binary matrices

Cf. A085659, A080858, A083029.

Cf. A085656, A114601.

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = if(k>n, r++; return); forvec(x=vector(k-1,i,[0,1]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Max Alekseyev)

More terms from Max Alekseyev, Dec 13 2005

A084552 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 7, 38, 286, 2686, 28512, 312572, 3337588, 40963216

Offset: 1

N. J. A. Sloane, Jul 13 2003

nonn

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).

This gives the number of different values of M + M' where M runs through the matrices counted in A127502.

			The singular matrix
2 -1 -1
-1 2 -1
-1 -1 2
is the only 3 X 3 matrix of this type which is not positive definite.

Index entries for sequences related to binary matrices

Cf. A084553, A085659, A080858, A083029.

a(6)-a(10) from Max Alekseyev, Jan 16 2006

A334399 Decimal expansion of sinh(e).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 26 2020

			(e^e - e^(-e))/2 = 7.54413710281697582634182004251653274...

Cf. A001113, A073226, A073230, A073742, A085659, A334400.

Mathematica
```
RealDigits[Sinh[E], 10, 110] [[1]]
```

Equals Sum_{k>=0} e^(2*k+1)/(2*k+1)!.

A366599 Decimal expansion of arcsinh(e).

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Oct 14 2023

			1.7253825588523150939450...

Cf. A001113, A073742, A085659, A188640, A334399, A365927.

Mathematica
```
RealDigits[ArcSinh[E], 10, 100] [[1]]
```

asinh(exp(1)) \\ Amiram Eldar, Oct 18 2023

Equals log(A188640). - Amiram Eldar, Oct 18 2023

A212436 Decimal expansion of the real part of e^(i/e).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 16 2012

Also cos(1/e).

The imaginary part of e^(i/e), or sin(1/e), is A212437.

			0.933092075598208563540410171408743589...

Cf. A212437, A085659, A085660.

RealDigits[Re[E^(I/E)],10,120][[1]] (* Harvey P. Dale, Sep 05 2022 *)

A212437 Decimal expansion of the imaginary part of e^(i/e).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 16 2012

Also sin(1/e).

The real part of e^(i/e), or cos(1/e) is in A212436.

			0.3596375654124955770382503939056247...

Cf. A212436, A085659, A085660.

RealDigits[Im[E^(I/E)],10,120][[1]] (* Harvey P. Dale, Jul 12 2021 *)

cp's OEIS Frontend

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

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A084552 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and -1's and 0's elsewhere.

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A334399 Decimal expansion of sinh(e).

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Mathematica

Formula

A366599 Decimal expansion of arcsinh(e).

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Mathematica

PARI

Formula

A212436 Decimal expansion of the real part of e^(i/e).

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Mathematica

A212437 Decimal expansion of the imaginary part of e^(i/e).

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Mathematica