|
%I #20 Feb 17 2022 01:04:36
%S 0,1,2,2,2,4,3,2,4,6,4,2,4,6,8,5,2,4,6,8,10,6,2,4,6,8,10,12,7,2,4,6,8,
%T 10,12,14,8,2,4,6,8,10,12,14,16,9,2,4,6,8,10,12,14,16,18,10,2,4,6,8,
%U 10,12,14,16,18,20,11,2,4,6,8,10,12,14,16,18,20,22,12,2,4,6,8,10,12,14,16
%N Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.
%C m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.
%H Vincenzo Librandi, <a href="/A056944/b056944.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 2n - floor((sqrt(8n+1)-1)/2)*ceiling((sqrt(8n+1)-1)/2) = 2n - A002024(n)*A003056(n) = 2n - A056942(n) = n -A056943(n). If n = t(t+1)/2 then a(n)=t; if n = t(t+1)/2+k with 0 < k <= t then a(n)=2k.
%e a(9)=6 since spiral is as marked by 9 X's in 4*3 = 12 rectangle, with 12-9 = 3 spaces unused and a used-unused difference of 9-3 = 6:
%e X.XX
%e X..X
%e XXXX
%e As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2. - _Gary W. Adamson_, Aug 26 2008
%t uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2},2n-Floor[c]Ceiling[c]]; Array[uar,90,0] (* _Harvey P. Dale_, Aug 14 2013 *)
%o (Magma) [2*n-Floor((Sqrt(8*n+1)-1)/2)*Ceiling((Sqrt(8*n+1)-1)/2): n in [0..90]]; // _Vincenzo Librandi_, Aug 06 2017
%Y Cf. A002024, A003056, A056942, A056943.
%Y Cf. A143595. - _Gary W. Adamson_, Aug 26 2008
%K easy,nonn,nice
%O 0,3
%A _Henry Bottomley_, Jul 13 2000
|
