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%I #53 Sep 08 2022 08:45:16
%S 1,6,16,30,49,72,100,132,169,210,256,306,361,420,484,552,625,702,784,
%T 870,961,1056,1156,1260,1369,1482,1600,1722,1849,1980,2116,2256,2401,
%U 2550,2704,2862,3025,3192,3364,3540,3721,3906,4096,4290,4489,4692,4900
%N Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).
%C A floretion-generated sequence.
%C 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
%C 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
%C Second bisection preceded by zero is A152743. - _Bruno Berselli_, Oct 25 2011
%C a(n) has no final digit 3, 7, 8. - _Paul Curtz_, Mar 04 2020
%C One odd followed by three evens.
%C From _Paul Curtz_, Mar 06 2020: (Start)
%C b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).
%C ( 25, 12, 4, 0, 1, 6, 16, 30, ...
%C -13, -8, -4 1, 5, 10, 14, 19, ...
%C 5, 4, 5, 4, 5, 4, 5, 4, ... .)
%C b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
%C A154589(n) are in the main diagonal of b(n) and b(-n). (End)
%H Vincenzo Librandi, <a href="/A102214/b102214.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
%F a(2n) = A016778(n) = (3n+1)^2.
%F a(n) + a(n+1) = A038764(n+1).
%F a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
%F a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - _Bruno Berselli_, Oct 25 2011
%F a(n) = A198392(n) + A198392(n-1). - _Bruno Berselli_, Nov 06 2011
%F From _Paul Curtz_, Mar 04 2020: (Start)
%F a(n) = A006578(n) + A001859(n) + A077043(n+1).
%F a(n) = A274221(2+2*n).
%F a(20+n) - a(n) = 30*(32+3*n).
%F a(1+2*n) = 3*(1+n)*(2+3*n).
%F a(n) = A047237(n) * A047251(n).
%F a(n) = A001651(n+1) * A032766(n).(End)
%F 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
%t 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 *)
%o (Magma) [(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // _Vincenzo Librandi_, Oct 26 2011
%o (PARI) a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ _Charles R Greathouse IV_, Apr 16 2020
%Y Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.
%Y Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Feb 17 2005
%E Definition rewritten by _Bruno Berselli_, Oct 25 2011
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