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

%I A008795 #93 Aug 15 2025 07:36:44
%S A008795 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,
%T A008795 78,105,91,120,105,136,120,153,136,171,153,190,171,210,190,231,210,
%U A008795 253,231,276,253,300,276,325,300,351,325,378,351,406,378,435,406,465,435,496,465,528,496
%N A008795 Molien series for 3-dimensional representation of dihedral group D_6 of order 6.
%C A008795 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
%C A008795 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
%C A008795 From _Omar E. Pol_, Feb 05 2012: (Start)
%C A008795 Also terms of A000217 and A000217-shifted interleaved.
%C A008795 Also 0 together with this sequence give the first row of the square array A194801. (End)
%C A008795 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
%H A008795 Vincenzo Librandi, <a href="/A008795/b008795.txt">Table of n, a(n) for n = 0..10000</a>
%H A008795 Kival Ngaokrajang, <a href="/A008795/a008795.pdf">Illustration of initial terms</a>
%H A008795 Ira Rosenholtz, <a href="https://www.jstor.org/stable/2690803">Problem 1584</a>, Mathematics Magazine, Vol. 72 (1999), p. 408.
%H A008795 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A008795 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008795 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A008795 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
%F A008795 a(n) = binomial(n/2+2, 2) for n even, binomial((n+1)/2, 2) for n odd. - _Rob Pratt_, Jul 12 2004
%F A008795 From _Paul Barry_, Jul 29 2004: (Start)
%F A008795 a(n-2) interleaves n(n+1)/2 and n(n-1)/2.
%F A008795 G.f.: (1-x+x^2)/((1+x)^2*(1-x)^3).
%F A008795 a(n) = (2*n^2 + 6*n + 7 + 3*(2*n+3)*(-1)^n)/16. (End)
%F A008795 a(n) = n*(n+1)/2, n = +- 1, +- 2... - _Omar E. Pol_, Feb 05 2012
%F A008795 From _Michael Somos_, Feb 01 2018: (Start)
%F A008795 Euler transform of length 6 sequence [0, 3, 1, 0, 0, -1].
%F A008795 G.f.: (1 + x^3) / (1 - x^2)^3.
%F A008795 a(n) = a(-3-n) for all in Z. (End)
%p A008795 a:= n-> binomial(n/2+2-3*irem(n, 2)/2, 2):
%p A008795 seq(a(n), n=0..70); # _Muniru A Asiru_, Feb 01 2018
%t A008795 Table[If[EvenQ[n], Binomial[n/2+2, 2], Binomial[(n+1)/2, 2]], {n, 0, 70}]
%t A008795 CoefficientList[Series[(1+x^3)/(1-x^2)^3, {x, 0, 70}], x] (* _Robert G. Wilson v_, Feb 05 2012 *)
%t A008795 a[ n_]:= Binomial[ Quotient[n, 2] + 2 - Mod[n, 2], 2]; (* _Michael Somos_, Feb 01 2018 *)
%t A008795 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 *)
%t A008795 LinearRecurrence[{1,2,-2,-1,1}, {1,0,3,1,6}, 70] (* _Robert G. Wilson v_, Feb 01 2018 *)
%o A008795 (Magma) [(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16: n in [0..70] ]; // _Vincenzo Librandi_, Aug 21 2011
%o A008795 (PARI) a(n)=(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16 \\ _Charles R Greathouse IV_, Oct 22 2015
%o A008795 (PARI) {a(n) = binomial(n\2 + 2 - n%2, 2)}; /* _Michael Somos_, Feb 01 2018 */
%o A008795 (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
%o A008795 (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
%Y A008795 Cf. A005044.
%Y A008795 First differences of A053307.
%K A008795 nonn,easy
%O A008795 0,3
%A A008795 _N. J. A. Sloane_
%E A008795 Definition clarified by _N. J. A. Sloane_, Feb 02 2018