A003947 Expansion of (1+x)/(1-4*x).

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080, 351843720888320

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Coordination sequence for infinite tree with valency 5.

For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 10 2007

Number of length-n strings of 5 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012

Create a rectangular prism with edges of lengths 2^(n-2), 2^(n-1), and 2^(n) starting at n=2; then the surface area = a(n). - J. M. Bergot, Aug 08 2013

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 306
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (4).
• Index entries for sequences related to trees

Crossrefs

Cf. A003948, A003949. Column 5 in A265583.

Programs

GAP

Concatenation([1], List([1..30], n-> 5*4^(n-1) )); # G. C. Greubel, Aug 10 2019

Magma

[1] cat [5*4^(n-1): n in [1..30]]; // G. C. Greubel, Aug 10 2019

Maple

k := 5; if n = 0 then 1 else k*(k-1)^(n-1); fi;

Mathematica

q = 5; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 5*4^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
LinearRecurrence[{4},{1,5},30] (* Harvey P. Dale, Apr 19 2015 *)

PARI

a(n)=5*4^n\4 \\ Charles R Greathouse IV, Sep 08 2011

Sage

[1]+[5*4^(n-1) for n in (1..30)] # G. C. Greubel, Aug 10 2019

Formula

Binomial transform of A060925. Its binomial transform is A003463 (without leading zero). - Paul Barry, May 19 2003

From Paul Barry, May 19 2003: (Start)

a(n) = (5*4^n - 0^n)/4.

G.f.: (1+x)/(1-4*x).

E.g.f.: (5*exp(4*x) - exp(0))/4. (End)

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 3. - Philippe Deléham, Jul 10 2005

a(n) = A146523(n)*A011782(n). - R. J. Mathar, Jul 08 2009

a(n) = 5*A000302(n-1), n>0.

a(n) = 4*a(n-1), n>1. - Vincenzo Librandi, Dec 31 2010

G.f.: 2+x- 2/G(0), where G(k)= 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013

Extensions

Edited by N. J. A. Sloane, Dec 04 2009