A024325 - OEIS

%I #8 Jan 29 2022 12:18:11

%S 0,0,5,7,10,13,15,18,33,38,44,48,54,60,64,70,98,106,114,121,130,137,

%T 145,153,160,169,213,223,233,244,255,265,275,286,297,307,317,328,391,

%U 403,416,430,442,456,469,481,496,508,521,534,547,561,644,659,675,690,707,722,737,755

%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 = A001950 (upper Wythoff sequence).

%H G. C. Greubel, <a href="/A024325/b024325.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A001950(n-j+1).

%t A023531[n_] := SquaresR[1, 8n+9]/2;

%t a[n_]:= a[n]= Sum[A023531[j]*Floor[(n-j+1)*GoldenRatio^2], {j,Floor[(n+1)/2]}];

%t Table[a[n], {n, 80}] (* _G. C. Greubel_, Jan 28 2022 *)

%o (Magma)

%o A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;

%o A024325:= func< n | (&+[A023531(j)*Floor((n-j+1)*(3+Sqrt(5))/2): j in [1..Floor((n+1)/2)]]) >;

%o [A024325(n) : n in [1..80]]; // _G. C. Greubel_, Jan 28 2022

%o (Sage)

%o def A023531(n):

%o     if ((sqrt(8*n+9) -3)/2).is_integer(): return 1

%o     else: return 0

%o def A023325(n): return sum( A023531(j)*floor(((n-j+1)*(3+sqrt(5)))/2) for j in (1..((n+1)//2)) )

%o [A023325(n) for n in (1..80)] # _G. C. Greubel_, Jan 28 2022

%Y Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024326, A024327.

%Y Cf. A001950, A023531.

%K nonn

%O 1,3

%A _Clark Kimberling_