A024324 - OEIS

%I #20 Feb 06 2022 02:16:41

%S 0,0,3,4,6,8,9,11,20,23,27,29,33,37,39,43,60,65,70,74,80,84,89,94,98,

%T 104,131,137,143,150,157,163,169,176,183,189,195,202,241,248,256,265,

%U 272,281,289,296,306,313,321,329,337,346,397,406,416,425,436,445,454,466,474,484

%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = A000201 (lower Wythoff sequence).

%H G. C. Greubel, <a href="/A024324/b024324.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000201(n-j+1).

%t Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Floor[n*GoldenRatio^2]; n -= m++]; t, {n, 120}] (* _G. C. Greubel_, Jan 28 2022 *)

%o (Magma)

%o b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;

%o A024324:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;

%o [A024324(n) : n in [1..80]]; // _G. C. Greubel_, Jan 28 2022

%o (Sage)

%o def b(n,j): return floor( (n+1-j)*(1+sqrt(5))/2 ) if ((sqrt(8*j+9) -3)/2).is_integer() else 0

%o def A024324(n): return sum( b(n,k) for k in (1..((n+1)//2)) )

%o [A024324(n) for n in (1..80)] # _G. C. Greubel_, Jan 28 2022

%o (PARI) my(phi=quadgen(5)); a(n) = my(L=n>>1,m=2,ret=0); n--; while(n>L, ret += floor(n*phi); n-=(m++)); ret; \\ _Kevin Ryde_, Feb 03 2022

%Y Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024325, A024326, A024327.

%Y Cf. A000201, A023531.

%K nonn

%O 1,3

%A _Clark Kimberling_

%E a(62) corrected by _Sean A. Irvine_, Jun 27 2019