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

0,2

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

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

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 *)

Prog

(PARI) a(n)=5*4^n\4 \\ Charles R Greathouse IV, Sep 08 2011
(Magma) [1] cat [5*4^(n-1): n in [1..30]]; // G. C. Greubel, Aug 10 2019
(Sage) [1]+[5*4^(n-1) for n in (1..30)] # G. C. Greubel, Aug 10 2019
(GAP) Concatenation([1], List([1..30], n-> 5*4^(n-1) )); # G. C. Greubel, Aug 10 2019

Crossrefs

Cf. A003948, A003949. Column 5 in A265583.

Sequence in context: A170590 A170638 A170686 * A369837 A363508 A252698

Adjacent sequences: A003944 A003945 A003946 * A003948 A003949 A003950

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Extensions

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

Status

approved