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A154738 Decimal expansion of (log(640320^3 + 744)/Pi)^2.

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

Offset: 3

Kousaka Hideaki (kousaka(AT)d6.dion.ne.jp), Jan 14 2009

A well-known real number that is very close to the integer 163.

RealDigits[(Log[640320^3+744]/Pi)^2,10,120][[1]] (* Harvey P. Dale, Jan 06 2012 *)

More terms from Klaus Brockhaus, Jan 15 2009

A154769 Zeroless primes with consecutive digits (1,..,9) starting with 1.

Original entry on oeis.org

1234567891, 1234567891234567891234567891, 1234567891234567891234567891234567891234567891234567891234567891234567

Offset: 1

Kousaka Hideaki (kousaka(AT)d6.dion.ne.jp), Jan 15 2009

The next term has 4690 digits. - Harvey P. Dale, Sep 27 2017

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 1234567891

Select[Table[FromDigits[PadRight[{},n,Range[9]]],{n,100}],PrimeQ] (* Harvey P. Dale, Sep 27 2017 *)

Name changed by Arkadiusz Wesolowski, Aug 24 2011

A154760 Final digit of n!! (A006882).

Original entry on oeis.org

1, 1, 2, 3, 8, 5, 8, 5, 4, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0

Offset: 0

Kousaka Hideaki (kousaka(AT)d6.dion.ne.jp), Jan 15 2009

For n>8: a(n)=0 if n is even, a(n)=5 if n is odd. - Dmitry Kamenetsky, Jan 17 2009

			The final digit of n! is 1,2,6,4,0,0,0,0,0,0,0,0,0,...

Join[Mod[Range[0,8]!!,10],PadRight[{},120,{5,0}]] (* Harvey P. Dale, Dec 16 2023 *)

Offset corrected, more terms added by R. J. Mathar, Jan 18 2009

A154841 Decimal expansion of e^(Pi*sqrt(43)).

Original entry on oeis.org

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

Offset: 9

Kousaka Hideaki (kousaka(AT)d6.dion.ne.jp), Jan 16 2009

A real number that is very close to the integer 884736744 = 960^3 + 744.

			884736743.999777466034906661937462078585376847399127139160917514627834...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 225-226.

RealDigits[E^(Pi Sqrt[43]), 10, 110][[1]] (* Bruno Berselli, Mar 05 2015 *)

exp(sqrt(43)*Pi) \\ Charles R Greathouse IV, Dec 06 2016

More digits from R. J. Mathar, Jan 21 2009

A115595 The sequence 11,0,1,3333,2,3,5555,4,5,7777,6,7,9999,9,0,2222,1,2,4444,3,4,6666,5,6,8888,7,9,11 has three subsequences that have interesting patterns inside it. Namely, 11,0,(1),3333,2,(3),5555,4,(5),7777,6,(7),9999,9,(0),2222,1,(2),4444,3,(4),6666,5,(6),8888,7,(9),11.

Original entry on oeis.org

11, 0, 1, 3333, 2, 3, 5555, 4, 5, 7777, 6, 7, 9999, 9, 0, 2222, 1, 2, 4444, 3, 4, 6666, 5, 6, 8888, 7, 9, 11

Offset: 1

Satoshi Hashiba (fantasia_sato205(AT)kcc.zaq.ne.jp), Mar 10 2006

You can generate very interesting sequences by using periodic numbers and the square root.

Cf. A021895, A021085, A113675, A113694.

Sqrt[991199991199991199991199991199991199991199991199991199]

You can get this sequence by Sqrt[991199991199991199991199991199991199991199991199991199], where Sqrt is the square root. Or (991199991199991199991199991199991199991199991199991199)/9.

A100539 Number of reduced Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

Original entry on oeis.org

1, 1, 1, 7132, 31503556

Offset: 1

Toru Ito (t_ito(AT)mue.biglobe.ne.jp), Nov 28 2004

Latin 4-dimensional hypercubes (Latin polyhedra) are a generalization of Latin cube and Latin square.

T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese).

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.

Cf. A100540, A132205, A132296.

a(5) from Ian Wanless, May 01 2008

Edited by N. J. A. Sloane, Dec 06 2009 at the suggestion of Vladeta Jovovic

A100540 Total number of Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

Original entry on oeis.org

1, 2, 48, 36972288, 52260618977280

Offset: 1

Toru Ito (t_ito(AT)mue.biglobe.ne.jp), Nov 28 2004

T. Ito, Method, equipment,program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (in Japanese).

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.

Cf. A100539, A132205, A132206.

A row of the array in A249026.

a(5) from Ian Wanless, May 01 2008

Edited by N. J. A. Sloane, Dec 05 2009 at the suggestion of Vladeta Jovovic

A098843 Number of reduced Latin cubes of order n.

Original entry on oeis.org

1, 1, 1, 64, 40246, 95909896152

Offset: 1

N. J. A. Sloane, based on correspondence from Toru Ito (t_ito(AT)mue.biglobe.ne.jp), Nov 03 2004

There are at least two ways to define Latin cubes - see the Preece et al. paper. - Rosemary Bailey, Nov 03 2004

T. Ito, Method for producing Latin squares, Publication number JP2000-28510A, Japan Patent Office.
T. Ito, Method for producing Latin squares, JP3394467B, Patent abstracts of Japan,Japan Patent Office.
Jia, Xiong Wei and Qin, Zhong Ping, The number of Latin cubes and their isotopy classes, J. Huazhong Univ. Sci. Tech. 27 (1999), no. 11, 104-106. MathSciNet #MR1751724.

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.
Gary L. Mullen, and Robert E. Weber, Latin cubes of order <= 5, Discrete Math. 32 (1980), no. 3, 291-297. (Gives a(1)-a(5).)
D. A. Preece, S. C. Pearce and J. R. Kerr, Orthogonal designs for three-dimensional experiments, Biometrika 60 (1973), 349-358.

Cf. A098846 (isomorphism classes), A098679 (total number), A099321 (isotopy classes).

a(6) computed independently by Brendan McKay and Ian Wanless, Dec 17 2004

A098679 Total number of Latin cubes of order n.

Original entry on oeis.org

1, 2, 24, 55296, 2781803520, 994393803303936000

Offset: 1

N. J. A. Sloane, based on correspondence from Toru Ito (t_ito(AT)mue.biglobe.ne.jp), Nov 06 2004

There are at least two ways to define Latin cubes - see the Preece et al. paper. - Rosemary Bailey, Nov 03 2004

T. Ito, Method for producing Latin squares, Publication number JP2000-28510A, Japan Patent Office.
T. Ito, Method for producing Latin squares, JP3394467B, Patent abstracts of Japan, Japan Patent Office.
Jia, Xiong Wei and Qin, Zhong Ping, The number of Latin cubes and their isotopy classes, J. Huazhong Univ. Sci. Tech. 27 (1999), no. 11, 104-106. MathSciNet #MR1751724.

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.
Gary L. Mullen, and Robert E. Weber, Latin cubes of order <= 5, Discrete Math. 32 (1980), no. 3, 291-297. (Gives a(1)-a(5).)
D. A. Preece, S. C. Pearce and J. R. Kerr, Orthogonal designs for three-dimensional experiments, Biometrika 60 (1973), 349-358.

Cf. A098843, A098846, A099321; A002860 (Latin squares).

A row of the array in A249026.

a(n) = n!*(n-1)!*(n-1)!*A098843(n).

a(6) computed independently by Brendan McKay and Ian Wanless, Dec 17 2004

A092297 Number of ways of 3-coloring an annulus consisting of n zones joined like a pearl necklace.

Original entry on oeis.org

0, 6, 6, 18, 30, 66, 126, 258, 510, 1026, 2046, 4098, 8190, 16386, 32766, 65538, 131070, 262146, 524286, 1048578, 2097150, 4194306, 8388606, 16777218, 33554430, 67108866, 134217726, 268435458, 536870910, 1073741826, 2147483646

Offset: 1

S. B. Step (stepy(AT)vesta.ocn.ne.jp), Feb 06 2004

A circular domain means a domain between two concentric circles and it is divided into n parts by n boundary lines perpendicular to the circles. Both sides of a line must have different colors. How many ways of coloring are there?
a(n) is also the multiple of six that's nearest to 2^n. - David Eppstein, Aug 31 2010
a(n) apparently is the trace of the n-th power of the adjacency matrix of the complete 3-graph, a 3 X 3 matrix with diagonal elements all zero and off-diagonal all ones (cf. A001045). If so, a(n) is the number of closed walks on the graph of length n. - Tom Copeland, Nov 06 2012
For n >= 2, a(n) is the number of length n words on 3 letters with no two consecutive like letters including the first and the last. Cf. A218034. - Geoffrey Critzer, Apr 05 2014

			a(2)=6 because we can color one zone in 3 colors and the other in 2, so 2*3=6 in all.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. Böhmová, C. Dalfó, and C. Huemer, On cyclic Kautz digraphs, Preprint 2016.
Cristina Dalfó, From subKautz digraphs to cyclic Kautz digraphs, arXiv:1709.01882 [math.CO], 2017.
Cristina Dalfó, The spectra of subKautz and cyclic Kautz digraphs, Linear Algebra and its Applications, 531 (2017), p. 210-219.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 25.
Paul P. Martin and Siti Fatimah Zakaria, Zeros of the 4-state Potts model partition function for the square lattice revisited, J. Stat. Mech. 084003 (2019). eq. (7).
Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Column k=3 of A106512.

Cf. A001045.

[2^n+2*(-1)^n : n in [1..40]]; // Vincenzo Librandi, Sep 27 2011

nn=28;Drop[CoefficientList[Series[6x^2/(1+x)^2/(1-3x/(1+x)),{x,0,nn}],x],1] (* Geoffrey Critzer, Apr 05 2014 *)
a[ n_] := 2 (2^(n - 1) + (-1)^n); (* Michael Somos, Oct 25 2014 *)
a[ n_] := If[ n < 1, -(-2)^(n - 1) a[2 - n] , (-1)^n HypergeometricPFQ[ Table[ -2, {k, n}], Table[ 1, {k, n - 1}], 1]] (* Michael Somos, Oct 25 2014 *)
LinearRecurrence[{1,2},{0,6},40] (* Harvey P. Dale, May 21 2024 *)

{a(n) = 2 * (2^(n-1) - (-1)^n)}; /* Michael Somos, Oct 25 2014 */

a(n) = 2^n + 2*(-1)^n; recurrence a(1)=0, a(2)=6, a(n) = 2*a(n-2) + a(n-1).
O.g.f: -6*x^2/((1+x)*(2*x-1)) = -3 - 1/(2*x-1) + 2/(1+x). - R. J. Mathar, Dec 02 2007
a(n) = 6*A001045(n-1). - R. J. Mathar, Aug 30 2008
a(n) = (-1)^n * a(2-n) * 2^(n-1) for all n in Z. - Michael Somos, Oct 25 2014

cp's OEIS Frontend

User: _ne_

A154738 Decimal expansion of (log(640320^3 + 744)/Pi)^2.

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A154769 Zeroless primes with consecutive digits (1,..,9) starting with 1.

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A154760 Final digit of n!! (A006882).

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A154841 Decimal expansion of e^(Pi*sqrt(43)).

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A115595 The sequence 11,0,1,3333,2,3,5555,4,5,7777,6,7,9999,9,0,2222,1,2,4444,3,4,6666,5,6,8888,7,9,11 has three subsequences that have interesting patterns inside it. Namely, 11,0,(1),3333,2,(3),5555,4,(5),7777,6,(7),9999,9,(0),2222,1,(2),4444,3,(4),6666,5,(6),8888,7,(9),11.

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A100539 Number of reduced Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

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A100540 Total number of Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

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A098843 Number of reduced Latin cubes of order n.

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A098679 Total number of Latin cubes of order n.

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