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%I #9 Jan 30 2014 11:40:43
%S 2,3,4,4,5,6,6,8,7,10,8,9,12,10,14,11,12,16,13,18,14,20,15,16,22,17,
%T 24,18,19,26,20,28,21,22,30,23,32,24,34,25,26,36,27,38,28,29,40,30,42,
%U 31,44,32,33,46,34,48,35,36,50,37,52,38,54,39,40,56,41,58
%N Manhattan distances between n^2 and (n+1)^2 in a left-aligned triangle with next M natural numbers in row M: 1, 2 3, 4 5 6, 7 8 9 10, etc.
%C Triangle in which we find distances begins:
%C _1
%C _2 3
%C _4 5 6
%C _7 8 9 10
%C 11 12 13 14 15
%C 16 17 18 19 20 21
%C 22 23 24 25 26 27 28
%C 29 30 31 32 33 34 35 36
%C 37 38 39 40 41 42 43 44 45
%C Subsequence of terms such that a(m)>=a(m-1) and a(m)>=a(m+1) seems to be A005843 (even numbers) except first two terms, and if such a(m) are removed, the remainder seems to be A000027 (natural numbers) except 1:
%C 2, 3, *4*, 4, 5, *6*, 6, *8*, 7, *10*, 8, 9, *12*, 10, *14*, 11, 12, *16*, 13, *18*, 14, *20*, 15, ...
%o (Python)
%o import math
%o def getXY(n):
%o y = int(math.sqrt(n*2))
%o if n<=y*(y+1)/2: y-=1
%o x = n - y*(y+1)/2
%o return x, y
%o for n in range(1, 77):
%o ox, oy = getXY(n*n)
%o nx, ny = getXY((n+1)**2)
%o print str(abs(nx-ox)+abs(ny-oy))+',',
%Y Cf. A236345, A005843, A000027.
%K nonn,easy
%O 1,1
%A _Alex Ratushnyak_, Jan 23 2014
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