A111386 -id:A111386 - cp's OEIS frontend

A111387 A111386(n-1) concatenated with A111386(n+1) divided by A111386(n).

5, 63, 35, 63, 335, 603, 335, 6003, 3335, 6003, 33335, 60003, 33335, 600003, 333335, 6000003, 3333335, 6000003, 3333335, 60000003, 33333335, 600000003, 333333335, 600000003, 3333333335, 6000000003, 3333333335, 60000000003

Offset: 2

Amarnath Murthy, Nov 11 2005

			15/3 = 5, 315/5 = 63.

Cf. A111386.

More terms from R. J. Mathar, Aug 20 2007

More terms computed from A111386. Max Alekseyev, Oct 17 2008

A056487 a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.

Original entry on oeis.org

1, 3, 5, 15, 25, 75, 125, 375, 625, 1875, 3125, 9375, 15625, 46875, 78125, 234375, 390625, 1171875, 1953125, 5859375, 9765625, 29296875, 48828125, 146484375, 244140625, 732421875, 1220703125, 3662109375, 6103515625, 18310546875, 30517578125, 91552734375

Offset: 0

Marks R. Nester

Apparently identical to A111386! Is this a theorem? - Klaus Brockhaus, Jul 21 2009

For n > 1, number of necklaces with n-1 beads and 5 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (0,5).

Column 5 of A284855.

Cf. A029744, A038754, A056391, A056451, A087205, A087206, A111386.

[n le 2 select 2*n-1 else 5*Self(n-2): n in [1..28]]; // Bruno Berselli, Mar 24 2011

A056487:=n->3^((1-(-1)^n)/2)*5^((2*n+(-1)^n-1)/4): seq(A056487(n), n=0..40); # Wesley Ivan Hurt, Nov 24 2016

Table[3^((1 - (-1)^n)/2)*5^((2*n + (-1)^n - 1)/4), {n, 0, 30}] (* Wesley Ivan Hurt, Nov 24 2016 *)
CoefficientList[Series[(1 + 3 x)/(1 - 5 x^2), {x, 0, 31}], x] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{0, 5}, {1, 3}, 35] (* Vincenzo Librandi, Nov 25 2016 *)
k=5; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, -1, 30}] (* Robert A. Russell, Sep 21 2018 *)

a(n)=if(n%2,3,1)*5^(n\2) \\ Charles R Greathouse IV, Oct 07 2015

def A056487(n): return 5**(n>>1)*(3 if n&1 else 1) # Chai Wah Wu, Oct 27 2022

a(n+2) = 5*a(n), a(0)=1, a(2)=3.
Binomial transform of A087205. Binomial transform is A087206. - Paul Barry, Aug 25 2003
G.f.: (1+3*x)/(1-5*x^2); a(n) = 5^(n/2)(1/2 + 3*sqrt(5)/10 + (1/2 - 3*sqrt(5)/10)(-1)^n). - Paul Barry, Mar 19 2004
2nd inverse binomial transform of Fibonacci(3n+2). - Paul Barry, Apr 16 2004
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 3^((1 - (-1)^n)/2) * 5^((2*n + (-1)^n-1)/4). - Bruno Berselli, Mar 24 2011
a(n+1) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 2, where k=5 is the number of possible colors. - Robert A. Russell, Sep 22 2018
E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5). - Stefano Spezia, Jun 06 2023

Changed one 'even' to 'odd' in the definition. - R. J. Mathar, Oct 06 2010

A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

Original entry on oeis.org

3, 5, 15, 25, 75, 125, 375, 625, 1875, 3125, 9375, 15625, 46875, 78125, 234375, 390625, 1171875, 1953125, 5859375, 9765625, 29296875, 48828125, 146484375, 244140625, 732421875, 1220703125, 3662109375, 6103515625, 18310546875

Offset: 1

Klaus Brockhaus, Jul 21 2009

Binomial transform is A163062, second binomial transform is A163063, third binomial transform is A098648 without initial 1, fourth binomial transform is A163064, fifth binomial transform is A163065.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A056487, A098648, A163062, A163063, A163064, A163065, A111386.

[ n le 2 select 2*n+1 else 5*Self(n-2): n in [1..29] ];

CoefficientList[Series[x*(3 + 5*x)/(1 - 5*x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
LinearRecurrence[{0,5},{3,5},30] (* Harvey P. Dale, Aug 01 2021 *)

x='x+O('x^30); Vec(x*(3+5*x)/(1-5*x^2)) \\ G. C. Greubel, Dec 21 2017

a(n) = (2-(-1)^n)*5^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(3+5*x)/(1-5*x^2).
a(n) = A056487(n), n>=1.
E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5) - 1. - Stefano Spezia, Nov 19 2023

cp's OEIS Frontend

A111387 A111386(n-1) concatenated with A111386(n+1) divided by A111386(n).

Views

Author

Keywords

Examples

Crossrefs

Extensions

A056487 a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula