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A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

%I A141683 #35 Sep 08 2022 08:45:35
%S A141683 1,1,2,4,9,19,41,88,189,406,872,1873,4023,8641,18560,39865,85626,
%T A141683 183916,395033,848491,1822473,3914488,8407925,18059374,38789712,
%U A141683 83316385,178955183,384377665,825604416,1773314929,3808901426
%N A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.
%C A141683 Essentially the same as A141015. - _R. J. Mathar_, Sep 14 2008
%H A141683 G. C. Greubel, <a href="/A141683/b141683.txt">Table of n, a(n) for n = 1..1000</a>
%H A141683 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1).
%F A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1. [That is, b(n) = A000931(n+4) = A078027(n+6) = A134816(n) = A182097(n+1). - _Petros Hadjicostas_, Aug 09 2020]
%F A141683 From _Colin Barker_, Feb 01 2012: (Start)
%F A141683 a(n) = a(n-1) + 2*a(n-2) + a(n-3), n > 4.
%F A141683 G.f.: x*(1 - x^2 - x^3)/(1 - x - 2*x^2 - x^3). (End)
%F A141683 a(n) = A000930(2*n - 3) for n >= 3. - _Georg Fischer_, Mar 23 2019
%t A141683 (* b = A000931 *)
%t A141683 b[0]=0; b[1]=1; b[2]=1; b[n_]:= b[n]= b[n-2] + b[n-3];
%t A141683 a[1]=1; a[n_]:= a[n]= Sum[b[k]*a[n-k], {k,n-1}];
%t A141683 Table[a[n], {n, 35}]
%t A141683 (* or *)
%t A141683 LinearRecurrence[{1, 2, 1}, {1, 1, 2, 4}, 31] (* _Georg Fischer_, Mar 23 2019 *)
%o A141683 (PARI) x='x+O('x^35); Vec(x*(1-x^2-x^3)/(1-x-2*x^2-x^3)) \\ _G. C. Greubel_, Jun 05 2018
%o A141683 (Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x^2-x^3)/(1-x-2*x^2-x^3))); // _G. C. Greubel_, Jun 05 2018
%Y A141683 Subsequence of A000930.
%Y A141683 Cf. A000931, A001906, A078027, A134816, A141015, A182097.
%K A141683 nonn,easy
%O A141683 1,3
%A A141683 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 07 2008