%I A056944
%S A056944 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 A056944 10,12,14,8,2,4,6,8,10,12,14,16,9,2,4,6,8,10,12,14,16,18,10,2,4,6,8,10,
%U A056944 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 A056944 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 A056944 m (when n is m-th triangular number) followed by m even numbers from 
               2 through 2m.
%F A056944 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 A056944 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 A056944 X.XX
%e A056944 X..X
%e A056944 XXXX
%e A056944 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 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 26 2008]
%Y A056944 Cf. A002024, A003056, A056942, A056943.
%Y A056944 A143595 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]
%Y A056944 Sequence in context: A064025 A054709 A121806 this_sequence A050493 A085454 
               A083403
%Y A056944 Adjacent sequences: A056941 A056942 A056943 this_sequence A056945 A056946 
               A056947
%K A056944 easy,nonn,nice
%O A056944 0,3
%A A056944 Henry Bottomley (se16(AT)btinternet.com), Jul 13 2000