A390661 - OEIS

%I #15 Jan 13 2026 08:08:59

%S 1,3,7,15,31,69,157,357,805,1811,4079,9199,20751,46797,105517,237917,

%T 536477,1209723,2727839,6151031,13869991,31275569,70523657,159024657,

%U 358586513,808580683,1823277559,4111329079,9270682359,20904566081,47137941057,106291873089

%N G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x^4))).

%H Vincenzo Librandi, <a href="/A390661/b390661.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 1 + 2 * Sum_{k=0..floor((n-1)/4)} a(k) * a(n-1-4*k).

%t a[0]=1; a[n_]:=a[n]=1+2*Sum[a[k]*a[n-1-4 k],{k,0,Floor[(n-1)/4]}]

%t Table[a[n],{n,0,30}] (* _Vincenzo Librandi_, Jan 12 2026 *)

%o (PARI) a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+2*sum(j=0, (i-1)\4, v[j+1]*v[i-4*j])); v;

%o (Magma) a := [1]; for n in [1..30] do s := 0; for k in [0..Floor((n-1)/4)] do s +:= a[k+1] * a[n - 4*k]; end for; Append(~a, 1 + 2*s); end for; a; // _Vincenzo Librandi_, Jan 12 2026

%Y Cf. A162326, A390659, A390660.

%Y Cf. A367692.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 14 2025