A008795 Molien series for 3-dimensional representation of dihedral group D_6 of order 6.

1, 0, 3, 1, 6, 3, 10, 6, 15, 10, 21, 15, 28, 21, 36, 28, 45, 36, 55, 45, 66, 55, 78, 66, 91, 78, 105, 91, 120, 105, 136, 120, 153, 136, 171, 153, 190, 171, 210, 190, 231, 210, 253, 231, 276, 253, 300, 276, 325, 300, 351, 325, 378, 351, 406, 378, 435, 406, 465, 435, 496, 465, 528, 496

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n-3) is the number of ordered triples of positive integers which are the side lengths of a nondegenerate triangle of perimeter n. - Rob Pratt, Jul 12 2004

a(n) is the number of ways to distribute n identical objects into 3 distinguishable bins so that no bin contains an absolute majority of objects. - Geoffrey Critzer, Mar 17 2010

From Omar E. Pol, Feb 05 2012: (Start)

Also terms of A000217 and A000217-shifted interleaved.

Also 0 together with this sequence give the first row of the square array A194801. (End)

a(n) is the number of coins left after packing 3-curves coins patterns into fountain of coins base n. Refer to A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". See illustration in links. - Kival Ngaokrajang, Oct 12 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Kival Ngaokrajang, Illustration of initial terms
• Ira Rosenholtz, Problem 1584, Mathematics Magazine, Vol. 72 (1999), p. 408.
• Index entries for sequences related to groups
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A005044.

First differences of A053307.

Programs

GAP

a := [1,0,3,1,6];; for n in [6..70] do a[n] := a[n-1] + 2*a[n-2] -2*a[n-3] -a[n-4] +a[n-5]; od; a; # Muniru A Asiru, Feb 01 2018

Magma

[(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16: n in [0..70] ]; // Vincenzo Librandi, Aug 21 2011

Maple

a:= n-> binomial(n/2+2-3*irem(n, 2)/2, 2):
seq(a(n), n=0..70); # Muniru A Asiru, Feb 01 2018

Mathematica

Table[If[EvenQ[n], Binomial[n/2+2, 2], Binomial[(n+1)/2, 2]], {n, 0, 70}]
CoefficientList[Series[(1+x^3)/(1-x^2)^3, {x, 0, 70}], x] (* Robert G. Wilson v, Feb 05 2012 *)
a[ n_]:= Binomial[ Quotient[n, 2] + 2 - Mod[n, 2], 2]; (* Michael Somos, Feb 01 2018 *)
a[ n_]:= With[ {m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ (1 - x + x^2) / ((1 - x)^3 (1 + x)^2), {x, 0, m}]]; (* Michael Somos, Feb 01 2018 *)
LinearRecurrence[{1,2,-2,-1,1}, {1,0,3,1,6}, 70] (* Robert G. Wilson v, Feb 01 2018 *)

PARI

a(n)=(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16 \\ Charles R Greathouse IV, Oct 22 2015

PARI

{a(n) = binomial(n\2 + 2 - n%2, 2)}; /* Michael Somos, Feb 01 2018 */

Sage

[(2*n^2 +6*n +7 +3*(2*n+3)*(-1)^n)/16 for n in (0..70)] # G. C. Greubel, Sep 11 2019

Formula

The signed version with g.f. (1-x^3)/(1-x^2)^3 is the inverse binomial transform of A084861. - Paul Barry, Jun 12 2003

a(n) = binomial(n/2+2, 2) for n even, binomial((n+1)/2, 2) for n odd. - Rob Pratt, Jul 12 2004

From Paul Barry, Jul 29 2004: (Start)

a(n-2) interleaves n(n+1)/2 and n(n-1)/2.

G.f.: (1-x+x^2)/((1+x)^2*(1-x)^3).

a(n) = (2*n^2 + 6*n + 7 + 3*(2*n+3)*(-1)^n)/16. (End)

a(n) = n*(n+1)/2, n = +- 1, +- 2... - Omar E. Pol, Feb 05 2012

From Michael Somos, Feb 01 2018: (Start)

Euler transform of length 6 sequence [0, 3, 1, 0, 0, -1].

G.f.: (1 + x^3) / (1 - x^2)^3.

a(n) = a(-3-n) for all in Z. (End)

Extensions

Definition clarified by N. J. A. Sloane, Feb 02 2018