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The trisection A178242(3n+2).
1

%I #32 Aug 12 2022 09:22:18

%S 7,25,26,44,133,187,125,161,403,493,296,350,817,943,539,611,1375,1537,

%T 854,944,2077,2275,1241,1349,2923,3157,1700,1826,3913,4183,2231,2375,

%U 5047,5353,2834,2996,6325,6667,3509,3689,7747,8125,4256,4454,9313,9727,5075,5291

%N The trisection A178242(3n+2).

%C For n = 0, 1, 2, 3, 4, 5, 6, 7, ...,

%C a(n-1) = -1, 7, 25, 26, 44, 133, 187, 125, ...

%C + A177049(n) = 1, 5, 14, 55, 91, 68, 95, 253, ...

%C gives 0, 12, 39, 81, 135, 201, 282, 378, ...

%C which are increasing multiples of 3.

%C a(n) mod 9 = period 4: repeat 7,7,8,8.

%H G. C. Greubel, <a href="/A178370/b178370.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).

%F a(n) = A060819(2+3*n)*(A060819(7+3*n) + A176672(n+2))/2. - corrected by _G. C. Greubel_, Feb 26 2020

%F G.f.: (7 +4*x -7*x^2 +46*x^3 -9*x^4 +8*x^5 +4*x^6 +2*x^7 -x^8)/((1-x)^3 * (1 + x^2)^3 ). - _R. J. Mathar_, Jan 16 2011

%F From _G. C. Greubel_, Feb 26 2020: (Start)

%F a(n) = (6 + i^n*(1 - i + (-1)^n*(1 + i)))*(9*n^2 + 27*n + 14)/16.

%F E.g.f.: ( 3*(14+36*x+9*x^2)*exp(x) + (14+36*x-9*x^2)*cos(x) + (14-36*x-9*x^2)*sin(x) )/8. (End)

%F Sum_{n>=0} 1/a(n) = 1 - (3 + 4*sqrt(3))*Pi/45. - _Amiram Eldar_, Aug 12 2022

%p m:=50; S:=series((7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 26 2020

%t LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1}, {7,25,26,44,133,187,125,161,403}, 50] (* _Harvey P. Dale_, May 21 2015 *)

%o (PARI) Vec( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) +O('x^50) ) \\ _G. C. Greubel_, Feb 26 2020

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) )); // _G. C. Greubel_, Feb 26 2020

%o (Sage)

%o def A178370_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) ).list()

%o A178370_list(50) # _G. C. Greubel_, Feb 26 2020

%Y Cf. A060819, A176672, A177049.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Dec 21 2010

%E More terms from _Jinyuan Wang_, Feb 26 2020