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A102214 Expansion of (1+4*x+4*x^2)/((1+x)*(1-x)^3). 4
1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A floretion-generated sequence.

a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006

Number of different partitions of numbers x+y=z such that {x,y,z} are integers {1,2,3,...,3n} and z>y>x [From Artur Jasinski, Feb 09 2010]

Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

G.f.: -(4*x^2+4*x+1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).

a(2n) = A016778(n) = (3n+1)^2.

a(n)+a(n+1) = A038764(n+1).

a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006

a(n) = (6*n*(3*n+4)+(-1)^n+7)/8.  - Bruno Berselli, Oct 25 2011

a(n) = A198392(n)+A198392(n-1).  - Bruno Berselli, Nov 06 2011

MATHEMATICA

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (*Artur Jasinski*) [From Artur Jasinski, Feb 09 2010]

PROG

(MAGMA) [(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

CROSSREFS

Cf. A016778, A038764.

Sequence in context: A168472 A054000 A113742 * A115007 A005891 A108182

Adjacent sequences:  A102211 A102212 A102213 * A102215 A102216 A102217

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Feb 17 2005

EXTENSIONS

Definition rewritten by Bruno Berselli, Oct 25 2011

STATUS

approved

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Last modified March 29 15:31 EDT 2017. Contains 284273 sequences.