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A102214
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Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).
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5
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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
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OFFSET
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0,2
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COMMENTS
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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. - Artur Jasinski, Feb 09 2010
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).
( 25, 12, 4, 0, 1, 6, 16, 30, ...
-13, -8, -4 1, 5, 10, 14, 19, ...
5, 4, 5, 4, 5, 4, 5, 4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)
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LINKS
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FORMULA
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G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020
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MATHEMATICA
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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, Feb 09 2010 *)
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PROG
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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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