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A176271
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The odd numbers as a triangle read by rows.
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24
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1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131
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OFFSET
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1,2
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COMMENTS
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T(n,k) = A005408(n*(n-1)/2 + k - 1);
Nicomachus: row sums give A000578;
A000537(n) = sum of first n rows;
ABS(alternating row sums) give A065599;
central terms give A016754: T(2*n-1,n) = A016754(n-1);
T(2*n,n) = A000466(n); T(2*n,n+1) = A053755(n);
T(n,k) + T(n,n-k+1) = A001105(n), 1 <= k <= n;
T(n,1) = A002061(n), central polygonal numbers;
T(n,2) = A027688(n-1) for n > 1;
T(n,3) = A027690(n-1) for n > 2;
T(n,4) = A027692(n-1) for n > 3;
T(n,5) = A027694(n-1) for n > 4;
T(n,6) = A048058(n-1) for n > 5;
T(n,n-3) = A108195(n-2) for n > 3;
T(n,n-2) = A082111(n-2) for n > 2;
T(n,n-1) = A014209(n-1) for n > 1;
T(n,n) = A028387(n-1);
A108309(n) = number of primes in n-th row.
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LINKS
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Table of n, a(n) for n=1..66.
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa
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FORMULA
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T(n,k) = n^2 - n + 2*k - 1, 1 <= k <= n.
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EXAMPLE
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From Philippe Deléham, Oct 03 2011: (Start)
Triangle begins:
1;
3, 5;
7, 9, 11;
13, 15, 17, 19;
21, 23, 25, 27, 29;
31, 33, 35, 37, 39, 41;
43, 45, 47, 49, 51, 53, 55;
57, 59, 61, 63, 65, 67, 69, 71;
73, 75, 77, 79, 81, 83, 85, 87, 89; (End)
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MAPLE
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A176271 := proc(n, k)
n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013
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PROG
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(Haskell)
a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012
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CROSSREFS
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Cf. A214604, A214661.
Sequence in context: A317439 A004273 A005408 * A144396 A060747 A089684
Adjacent sequences: A176268 A176269 A176270 * A176272 A176273 A176274
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KEYWORD
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nonn,tabl
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AUTHOR
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Reinhard Zumkeller, Apr 13 2010
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STATUS
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approved
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