

A008795


Molien series for 3dimensional group [2,n]+ = 22n.


15



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
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OFFSET

0,3


COMMENTS

a(n3) = 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
Contribution from Omar E. Pol, Feb 05 2012 (Start:)
Also positive terms of A000217 and A000217 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 3curves 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


REFERENCES

Ira Rosenholtz, Problem 1584, Mathematics Magazine, Vol. 72 (1999), p. 408.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Kival Ngaokrajang, Illustration of initial terms
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).


FORMULA

The signed version with g.f. (1x^3)/(1x^2)^3 is the inverse binomial transform of A084861.  Paul Barry, Jun 12 2003
a(n) = binom(n/2+2, 2) for n even, binom((n+1)/2, 2) for n odd  Rob Pratt, Jul 12 2004
a(n2) interleaves n(n+1)/2 and n(n1)/2. G.f.: (1x+x^2)/((1+x)^2(1x)^3)); a(n)=(2n^2+6n+7)/16+3(2n+3)(1)^n/16.  Paul Barry, Jul 29 2004
a(n) = n*(n+1)/2, n = + 1, + 2...  Omar E. Pol, Feb 05 2012


MATHEMATICA

Table[If[EvenQ[n], Binomial[n/2+2, 2], Binomial[(n+1)/2, 2]], {n, 0, 49}]
CoefficientList[ Series[(1 + x^3)/(1  x^2)^3, {x, 0, 50}], x] (* Robert G. Wilson v, Feb 05 2012 *)


PROG

(MAGMA) [(2*n^2+6*n+7)/16+3*(2*n+3)*(1)^n/16: n in [0..60] ]; // Vincenzo Librandi, Aug 21 2011
(PARI) a(n)=(2*n^2+6*n+7)/16+3*(2*n+3)*(1)^n/16 \\ Charles R Greathouse IV, Oct 22 2015


CROSSREFS

Cf. A005044.
First differences of A053307.
Sequence in context: A158822 A226132 A121443 * A165188 A132180 A207630
Adjacent sequences: A008792 A008793 A008794 * A008796 A008797 A008798


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



