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A287548
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Triangle read by rows: T(n,k), where each row begins with the Catalan number for n nonintersecting arches and transitions through k generations of eliminating and reducing arch configurations to an end row entry equal to number of semi-meander solutions for n arches.
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1
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1, 2, 1, 5, 3, 2, 14, 9, 7, 4, 42, 28, 23, 16, 10, 132, 90, 76, 57, 42, 24, 429, 297, 255, 199, 156, 108, 66, 1430, 1001, 869, 695, 563, 420, 304, 174, 4862, 3432, 3003, 2442, 2019, 1568, 1210, 836, 504
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OFFSET
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1,2
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LINKS
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FORMULA
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T(n,1) = Catalan Numbers C(n)= A000108(n).
Conjectured:
T(n,2) = C(n) - C(n-1) = A000245(n-1).
T(n,3) = C(n) - C(n-1) - C(n-2) = A067324(n-3).
T(n,4) = C(n) - C(n-1) - 2*C(n-2) - C(n-3).
T(n,n) = semi-meander solutions = A000682(n-1).
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EXAMPLE
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Triangle begins:
n\k 1 2 3 4 5 6 7 8
1: 1
2: 2 1
3: 5 3 2
4: 14 9 7 4
5: 42 28 23 16 10
6: 132 90 76 57 42 24
7: 429 297 255 199 156 108 66
8: 1430 1001 869 695 563 420 304 174
...
Capital letters (U,D) represent beginning and end of first and last arch. Only 1 UD ends arch sequence in next generation.
Reduction of arches: Elimination of arches:
(middle D U = new arch U D in the next arch generation)
/\
/\ //\\ /\/\/\/\ = UDududUD
//\\/\///\\\ = UudDudUuuddD /\
/\ /\ / \
/\//\\//\\ = UDuuddUudD //\/\\ = UududD
end
For n=3 C(n)=5 nonintersecting arch configurations:
UuuddD UududD UudDUD UDUudD UDudUD T(3,1)=5
end end UDUD UDUD UudD T(3,2)=3
UD UD end T(3,3)=2
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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