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

1, 3, 3, 7, 9, 15, 22, 33, 48, 71, 101, 147, 208, 297, 419, 591, 829, 1161, 1619, 2255, 3130, 4339, 6000, 8285, 11419, 15717, 21600, 29649, 40645, 55659, 76135, 104043, 142045, 193759, 264078, 359637, 489408, 665539, 904449, 1228343, 1667216, 2261593, 3066183

Offset: 0

Author

N. J. A. Sloane, Apr 23 2017

Keywords

• nonn

Links

• Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276. See Prop. 7.1.
• Index entries for linear recurrences with constant coefficients, signature (0,3,2,-3,-4,0,2,1).

Crossrefs

Cf. A000931, A284709.

Programs

Maple

A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else procname(n-2)+procname(n-3); fi; end;
psi:=n->A000931(n+6);
f:=n->add(psi(k-1)*psi(n-k-1),k=0..n)+(1-(-1)^n)/2;
[seq(f(n),n=0..40)];

Mathematica

(* b is A000931 *)
b[n_] := b[n] = Which[n == 0, 1, n <= 2, 0, True, b[n-2] + b[n-3]];
psi[n_] := b[n+6];
a[n_] := Sum[psi[k-1]*psi[n-k-1], {k, 0, n}] + (1-(-1)^n)/2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 19 2023, after Maple code *)

Formula

From Colin Barker, Apr 25 2017: (Start)

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).

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)