login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A089977 Expansion of 1/((1-2*x)*(1+x+2*x^2)). 6
1, 1, 1, 5, 9, 13, 33, 69, 121, 253, 529, 1013, 2025, 4141, 8193, 16293, 32857, 65629, 130801, 262229, 524745, 1047949, 2096865, 4195845, 8387641, 16775101, 33558481, 67109045, 134209449, 268443373, 536879553, 1073717349, 2147490841 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums of the Riordan array (1,x(1+4x^2)). - Paul Barry, Jan 12 2006

6*a(n-3) is the number of distinct nonbacktracking paths of length n on a unit cube which start on a given vertex and end on the same one (if n is even) or the opposite one (if n is odd). E.g., a(7)=69 because a(7)=a(6)+4*a(4)=33+4*9=69. a(3)=5 because there are 6*a(6-3)=6*5=30 nonbacktracking paths of length 6 on a unit cube that end on the same vertex (6 is even); if we name the vertices of a unit cube ABCDEFGH in the order of x+2y+4z, such paths starting from A are ABDCGEA, ABDHFBA, ABDHFEA, ABDHGCA, ABDHGDA; the remaining 25 can be derived from these 5 reflecting them about the ABGH plane and rotating the resulting 10 around the AH axis by 120 and -120 degrees. - Michal Kaczmarczyk, Apr 24 2006

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n.  For n>=3, 5*a(n-3) equals the number of 5-colored compositions of n with all parts >=3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n+2) equals the number of words of length n on alphabet {0,1,2,3,4}, having at least two zeros between every two successive nonzero letters. - Milan Janjic, Feb 07 2015

Number of compositions of n into one sort of part 1 and four sorts of part 3 (the g.f. is 1/(1-x-4*x^3) ). - Joerg Arndt, Feb 07 2015

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,4).

FORMULA

a(n) = Sum_{k=0..floor(n/2)} C(n-2*k, k)*4^k.

a(n) = 2^(n-1)+2^(n/2)*(cos((n+2)*arctan(sqrt(7)/7)+Pi*n/2)/4+5*sqrt(7)*sin((n+2)*arctan(sqrt(7)/7)+Pi*n/2)/28).

a(n) = Sum_{k=0..n} C(k, floor((n-k)/2))2^(n-k)*(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006

a(n) = a(n-1) + 4*a(n-3) for n>=3, a(0)=1, a(1)=1, a(2)=1. - Michal Kaczmarczyk, Apr 24 2006

a(n) = 2^(n-1) + A110512(n)/2. - R. J. Mathar, Aug 23 2011

G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 4*x^2)/( x*(4*k+3 + 4*x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013

a(n) = hypergeom([1/3-n/3,2/3-n/3,-n/3],[1/2-n/2,-n/2],-27). - Peter Luschny, Feb 07 2015

MAPLE

seq(add(binomial(n-2*k, k)*4^k, k=0..floor(n/3)), n=0..32); # Zerinvary Lajos, Apr 03 2007

MATHEMATICA

Table[HypergeometricPFQ[{1/3-n/3, 2/3-n/3, -n/3}, {1/2-n/2, -n/2}, -27], {n, 0, 32}] (* Peter Luschny, Feb 07 2015 *)

CoefficientList[Series[1/((1 - 2*x)*(1 + x + 2*x^2)), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)

PROG

(PARI) Vec(1/((1-2*x)*(1+x+2*x^2)) + O(x^50)) \\ Michel Marcus, Feb 07 2015

CROSSREFS

Cf. A084386, A077949, A000930.

Sequence in context: A170896 A257171 A233973 * A024728 A024950 A180514

Adjacent sequences:  A089974 A089975 A089976 * A089978 A089979 A089980

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 18 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 21 06:31 EST 2017. Contains 294989 sequences.