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A008795
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Molien series for 3-dimensional representation of dihedral group D_6 of order 6.
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21
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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
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OFFSET
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0,3
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COMMENTS
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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
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
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LINKS
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Ira Rosenholtz, Problem 1584, Mathematics Magazine, Vol. 72 (1999), p. 408.
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FORMULA
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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
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
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)
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MAPLE
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a:= n-> binomial(n/2+2-3*irem(n, 2)/2, 2):
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MATHEMATICA
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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 *)
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PROG
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(Magma) [(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16: n in [0..70] ]; // Vincenzo Librandi, Aug 21 2011
(PARI) {a(n) = binomial(n\2 + 2 - n%2, 2)}; /* Michael Somos, Feb 01 2018 */
(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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