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%I #56 Apr 21 2023 18:10:05
%S 1,3,5,6,6,6,7,9,11,12,12,12,13,15,17,18,18,18,19,21,23,24,24,24,25,
%T 27,29,30,30,30,31,33,35,36,36,36,37,39,41,42,42,42,43,45,47,48,48,48,
%U 49,51,53,54,54,54,55,57,59,60,60,60,61,63,65,66,66,66,67,69,71,72,72,72,73,75
%N Expansion of g.f. (1 - x)^(-1)/(1 - 2*x + 2*x^2 + x^3).
%C Partial sums of A021823. Second partial sums of A010892. - _Paul Barry_, Jun 06 2003
%C Row sums of A144083. - _Gary W. Adamson_, Sep 10 2008
%H Vincenzo Librandi, <a href="/A077859/b077859.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,3,-1).
%F G.f.: 1/((1-x)^2*(1-x+x^2)).
%F a(n) = Sum_{k=0..n} (k+1)*2*sin(Pi(n-k)/3 + Pi/3)/sqrt(3). - _Paul Barry_, May 18 2004
%F a(n) = Sum_{k=0..n} binomial(n-2k, n-k-1). - _Paul Barry_, Jan 15 2005
%F a(n) = n + 2 + (-1 + n - 3*floor(n/3))*(-1)^floor(n/3). - _Tani Akinari_, Jun 27 2013
%F a(n) = n + 1 + a(n-1) - a(n-2), with a(-1) = a(-2) = 0. - _Richard R. Forberg_, Jul 11 2013
%F a(n) = 3*a(n-1) - 4*a(n-2) + 3*a(n-3) - 1*a(n-4). - _Joerg Arndt_, Jul 12 2013
%F a(n) = Sum_{k=0..n} (-1)^k*(n+1-k)*b(k), where b(n) = A049347(n). - _Mircea Merca_, Feb 04 2014
%F E.g.f.: exp(x)*(2 + x) + exp(x/2)*(sqrt(3)*sin(sqrt(3)*x/2) - 3*cos(sqrt(3)*x/2))/3. - _Stefano Spezia_, Feb 11 2023
%F Sum_{n>=0} (-1)^n/a(n) = log(2)/3 + log(3)/2 = log(108)/6. - _Amiram Eldar_, Feb 14 2023
%p A010892 := proc(n) op(1+(n mod 6),[1,1,0,-1,-1,0]) ; end proc:
%p A077859 := proc(n) n+2+A010892(n+4) ; end proc:
%p seq(A077859(n),n=0..50) ; # _R. J. Mathar_, Mar 22 2011
%t CoefficientList[Series[(1 - x)^(-1)/(1 - 2 x + 2 x^2 - x^3), {x, 0, 100}], x] (* _Vincenzo Librandi_, Feb 04 2014 *)
%t LinearRecurrence[{3,-4,3,-1},{1,3,5,6},80] (* _Harvey P. Dale_, Apr 21 2023 *)
%o (PARI) Vec(1/(1-x)/(1-2*x+2*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%o (Magma) I:=[1, 3, 5, 6]; [n le 4 select I[n] else 3*Self(n-1)-4*Self(n-2)+3*Self(n-3)-Self(n-4): n in [1..100]]
%Y Cf. A010892, A021823, A049347, A144083.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 17 2002
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