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A214871
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Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.
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1
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1, 3, 4, 6, 2, 7, 11, 8, 9, 12, 18, 13, 5, 14, 19, 27, 20, 15, 16, 21, 28, 38, 29, 22, 10, 23, 30, 39, 51, 40, 31, 24, 25, 32, 41, 52, 66, 53, 42, 33, 17, 34, 43, 54, 67, 83, 68, 55, 44, 35, 36, 45, 56, 69, 84, 102, 85, 70, 57, 46, 26, 47, 58, 71, 86, 103, 123
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OFFSET
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1,2
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COMMENTS
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a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
. . .
T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1);
. . .
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LINKS
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FORMULA
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As table
T(n,k) = (n-1)^2+1, if n=k;
T(n,k) = (n-1)^2+2*k+1, if n>k;
T(n,k) = (k-1)^2+2*n, if n<k.
As linear sequence
a(n) = (i-1)^2+1, if i=j;
a(n) = (i-1)^2+2*j+1, if i>j;
a(n) = (j-1)^2+2*i, if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
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EXAMPLE
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The start of the sequence as table:
1....3...6..11..18..27...
4....2...8..13..20..29...
7....9...5..15..22..31...
12..14..16..10..24..33...
19..21..23..25..17..35...
28..30..32..34..36..26...
. . .
The start of the sequence as triangle array read by rows:
1;
3,4;
6,2,7;
11,8,9,12;
18,13,5,14,19;
27,20,15,16,21,28;
. . .
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PROG
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(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i == j:
result=(i-1)**2+1
if i > j:
result=(i-1)**2+2*j+1
if i < j:
result=(j-1)**2+2*i
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CROSSREFS
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Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A059100, A087475, A114949, A189833, A114948, A114962; in columns A117950, A117951, A117619, A189834, A189836; the main diagonal is A002522.
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KEYWORD
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AUTHOR
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STATUS
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approved
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