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A285187 a(n) = Sum(psi(k-1)*psi(n-k-1),k=0..n)+(1-(-1)^n)/2, where psi(k) = A000931(k+6).

%I A285187 #25 Mar 19 2023 09:43:33
%S A285187 1,3,3,7,9,15,22,33,48,71,101,147,208,297,419,591,829,1161,1619,2255,
%T A285187 3130,4339,6000,8285,11419,15717,21600,29649,40645,55659,76135,104043,
%U A285187 142045,193759,264078,359637,489408,665539,904449,1228343,1667216,2261593,3066183
%N A285187 a(n) = Sum(psi(k-1)*psi(n-k-1),k=0..n)+(1-(-1)^n)/2, where psi(k) = A000931(k+6).
%H A285187 Tomislav Doslic and I. Zubac, <a href="https://doi.org/10.26493/1855-3974.851.167">Counting maximal matchings in linear polymers</a>, Ars Mathematica Contemporanea 11 (2016) 255-276. See Prop. 7.1.
%H A285187 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,2,-3,-4,0,2,1).
%F A285187 From _Colin Barker_, Apr 25 2017: (Start)
%F A285187 G.f.: (1 + 3*x - 4*x^3 - 3*x^4 + x^5 + 2*x^6 + x^7) / ((1 - x)*(1 + x)*(1 - x^2 - x^3)^2).
%F A285187 a(n) = 3*a(n-2) + 2*a(n-3) - 3*a(n-4) - 4*a(n-5) + 2*a(n-7) + a(n-8) for n>7. (End)
%p A285187 A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else procname(n-2)+procname(n-3); fi; end;
%p A285187 psi:=n->A000931(n+6);
%p A285187 f:=n->add(psi(k-1)*psi(n-k-1),k=0..n)+(1-(-1)^n)/2;
%p A285187 [seq(f(n),n=0..40)];
%t A285187 (* b is A000931 *)
%t A285187 b[n_] := b[n] = Which[n == 0, 1, n <= 2, 0, True, b[n-2] + b[n-3]];
%t A285187 psi[n_] := b[n+6];
%t A285187 a[n_] := Sum[psi[k-1]*psi[n-k-1], {k, 0, n}] + (1-(-1)^n)/2;
%t A285187 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 19 2023, after Maple code *)
%Y A285187 Cf. A000931, A284709.
%K A285187 nonn
%O A285187 0,2
%A A285187 _N. J. A. Sloane_, Apr 23 2017