A025096 - OEIS

%I #7 Sep 09 2022 03:55:09

%S 0,0,1,3,4,0,0,0,1,3,4,7,11,18,29,47,76,0,1,3,4,7,11,18,29,47,76,123,

%T 199,322,521,843,1364,2208,3574,5782,9356,11,18,29,47,76,123,199,322,

%U 521,843,1364,2207,3571,5778,9349,15127,24476,39604,64082,103686,167768,271454

%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A000032, t = A023533.

%H G. C. Greubel, <a href="/A025096/b025096.txt">Table of n, a(n) for n = 2..5000</a>

%t A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2,3]!= n,0,1];

%t A025096[n_]:= A025096[n]= Sum[LucasL[j]*A023533[n-j+1], {j, Floor[n/2]}];

%t Table[A025096[n], {n,2,100}] (* _G. C. Greubel_, Sep 08 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o A025096:= func< n | (&+[Lucas(k)*A023533(n+1-k): k in [1..Floor(n/2)]]) >;

%o [A025096(n): n in [2..130]]; // _G. C. Greubel_, Sep 08 2022

%o (SageMath)

%o @CachedFunction

%o def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1

%o def A025096(n): return sum(lucas_number2(k,1,-1)*A023533(n-k+1) for k in (1..(n//2)))

%o [A025096(n) for n in (2..100)] # _G. C. Greubel_, Sep 08 2022

%Y Cf. A000032, A023533.

%K nonn

%O 2,4

%A _Clark Kimberling_

%E Offset corrected by _G. C. Greubel_, Sep 08 2022