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A347433
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Irregular triangle read by rows: T(n,k) is the difference between the total arch lengths of a semi-meander multiplied by its number of exterior arches and total arch lengths of the semi-meanders with n + 1 top arches generated by the exterior arch splitting algorithm on the given semi-meander.
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0
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4, 4, 4, 10, 4, 11, 4, 12, 20, 4, 13, 22, 4, 14, 24, 34, 4, 15, 26, 37, 4, 16, 28, 40, 52, 4, 17, 30, 43, 56, 4, 18, 32, 46, 60, 74, 4, 19, 34, 49, 64, 79, 4, 20, 36, 52, 68, 84, 100, 4, 21, 38, 55, 72, 89, 106, 4, 22, 40, 58, 76, 94, 112, 130, 4, 23, 42, 61, 80, 99
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OFFSET
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2,1
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LINKS
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FORMULA
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For n >= 2 and k = 2..floor((n+2)/2), T(n,k) = 4 + (n+2)*(k-2).
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EXAMPLE
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n = number of top arches, k = number of exterior top arches:
n\k 2 3 4 5 6
2: 4
3: 4
4: 4 10
5: 4 11
6: 4 12 20
7: 4 13 22
8: 4 14 24 34
9: 4 15 26 37
10: 4 16 28 40 52
Length of each arch = 1 + number of arches covered:
Top arches of a given semi-meander: Arch splitting generated
n = 5, k = 2 semi-meanders (6 top arches):
1 1 = 2 exterior arches /\
/\ //\\
/\ //\\ ///\\\
//\\ ///\\\ /\ /\ ////\\\\
21 321 = 9 length of top arches 1 1 4321 = 12 length of top arches
/\
//\\ /\
///\\\ //\\ /\
321 21 1 = 10 length of top arches
T(5,2) = 4 + (5+2)(2-2) = 4 --------------------------- 4 = (12+10) - (2 * 9);
Top arches of given semi meander:
n = 5, k = 3 /\
1 1 1 = 3 exterior arches / \
/\ /\ / \
/\ //\\ //\\ //\ /\\
1 21 21 = 7 length top arches /\ ///\\//\\\
1 521 21 = 12 length of top arches
/\
/\ //\\
//\\ /\ ///\\\
21 1 321 = 10 length of top arches
/\
/ \
/ /\\
//\//\\\ /\ /\
41 21 1 1 = 10 length of top arches
T(5,3) = 4 + (5+2)(3-2) = 11 --------------------- 11 = (12+10+10) - (3 * 7).
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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