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A008795 Molien series for 3-dimensional 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n-3) = 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 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

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. (1-x^3)/(1-x^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(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)=(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

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Last modified August 22 10:37 EDT 2017. Contains 290946 sequences.