A047240 -id:A047240 - cp's OEIS frontend

A109340 Expansion of x^2(1+x+4x^2)/((1+x+x^2)*(1-x)^3).

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

Offset: 0

Creighton Dement, Aug 20 2005

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.
Example:
Gear wheels (*) and free axles (·):
              ·
             * *
    *       * · *
   · *     · * * ·
  * * ·   * * · * *
   n=3       n=5
n=3: 1 half-hexagon, a(2)=1.
n=5: 3 half-hexagons and 1 full hexagon containing 6 half-hexagons -> a(4)=3+6*1=9.
See "Connected gear wheels" link.
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)
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)

Gerhard Kirchner, Connected gear wheels
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A105770, A047240, A056107.

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 *)

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(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=9. - Harvey P. Dale, Jun 24 2013
a(n) = (n-1)^2 - ((n+1) mod 3) mod 2, n >= 1. - Gerhard Kirchner, Jan 20 2017
E.g.f.: (exp(x)*(2 + 3*(x - 1)*x) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 23 2022

A305859 Numbers that are congruent to {1, 3, 11} mod 12.

Original entry on oeis.org

1, 3, 11, 13, 15, 23, 25, 27, 35, 37, 39, 47, 49, 51, 59, 61, 63, 71, 73, 75, 83, 85, 87, 95, 97, 99, 107, 109, 111, 119, 121, 123, 131, 133, 135, 143, 145, 147, 155, 157, 159, 167, 169, 171, 179, 181, 183, 191, 193, 195, 203, 205, 207, 215, 217, 219, 227, 229, 231, 239

Offset: 1

Vincenzo Librandi, Jun 12 2018

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

Equals 2*A047240 - 1 and 2*A047266 + 1 (after 0).

[n: n in [0..300] | n mod 12 in [1,3,11]]; // Bruno Berselli, Jun 13 2018

Table[2 n + 6 Floor[n/3] - 1, {n, 1, 60}] (* Bruno Berselli, Jun 13 2018 *)
LinearRecurrence[{1,0,1,-1},{1,3,11,13},60] (* Harvey P. Dale, Mar 15 2023 *)

G.f.: x*(1 + 2*x + 8*x^2 + x^3)/((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>12.
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>6.
a(n) = 2*n + 6*floor(n/3) - 1. - Bruno Berselli, Jun 13 2018

Edited by Bruno Berselli, Jun 13 2018

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A109340 Expansion of x^2(1+x+4x^2)/((1+x+x^2)*(1-x)^3).

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A305859 Numbers that are congruent to {1, 3, 11} mod 12.

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cp's OEIS Frontend

A109340 Expansion of x^2*(1+x+4*x^2)/((1+x+x^2)*(1-x)^3).

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A305859 Numbers that are congruent to {1, 3, 11} mod 12.

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Magma

Mathematica

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Extensions

A109340 Expansion of x^2(1+x+4x^2)/((1+x+x^2)*(1-x)^3).