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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A109340 Expansion of x^2*(1+x+4*x^2)/((1+x+x^2)*(1-x)^3). 1
0, 0, 1, 3, 9, 16, 24, 36, 49, 63, 81, 100, 120, 144, 169, 195, 225, 256, 288, 324, 361, 399, 441, 484, 528, 576, 625, 675, 729, 784, 840, 900, 961, 1023, 1089, 1156, 1224, 1296, 1369, 1443, 1521, 1600, 1680, 1764, 1849, 1935, 2025, 2116, 2208, 2304, 2401 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Comment from Gerhard Kirchner, Jan 20 2017 (Start)

According to the game "Mecanix":

In a triangular arrangement of wheel axles (n rows with 1, 2,.., n axles), a connected set of unblocked gear wheels is installed such that the number of wheel quadruples forming half-hexagons is maximal.

a(n-1) is the maximum number. See example and link "Connected gear wheels".

Annotation: In such a configuration also the number of wheels is maximal. It is A007980(n). For n<3, however, there is no half-hexagon. (End)

LINKS

Table of n, a(n) for n=0..50.

Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Gerhard Kirchner, Connected gear wheels

FORMULA

a(n+1) - a(n) = A047240(n); a(n) + a(n+1) + a(n+2) = A056107(n); a(n+2) - a(n+1) + a(n) = A105770(n)

a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=9, a(n)=2*a(n-1)-a(n-2)+ a(n-3)- 2*a(n-4)+ a(n-5). - Harvey P. Dale, Jun 24 2013

a(n) = (n-1)^2 - [(n+1) mod 3] mod 2, n>=1 - Gerhard Kirchner, Jan 20 2017

EXAMPLE

Gear wheels (*) and free axles (·):

a             ·

b            * *

c   *       * · *

d  · *     · * * ·

e * * ·   * * · * *

   n=3       n=5

n=3: 1 half-hexagon, a(n-1)=1

n=5: 3 half-hexagons and 1 full hexagon containing 6 half- Hexagons -> a(n-1)=3+6*1=9

MATHEMATICA

CoefficientList[Series[x^2(1+x+4x^2)/((1+x+x^2)(1-x)^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 1, 3, 9}, 60] (* Harvey P. Dale, Jun 24 2013 *)

PROG

Floretion Algebra Multiplication Program, FAMP Code: 4tessumrokseq[A*B] with A = + .5'i + .5'j + .5'k + .5e and B = + .5i' + .5j' + .5k' + .5e; roktype: Y[15] = Y[15] + p; sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code)

CROSSREFS

Cf. A105770, A047240, A056107.

Sequence in context: A020967 A253547 A290371 * A024385 A212566 A061942

Adjacent sequences:  A109337 A109338 A109339 * A109341 A109342 A109343

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Aug 20 2005

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 October 18 16:30 EDT 2017. Contains 293524 sequences.