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A176271
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The odd numbers as a triangle read by rows.
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25
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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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A108309(n) = number of primes in n-th row.
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LINKS
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FORMULA
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T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) =A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) =A016754(n-1) (main diagonal).
T(n, k) + T(n, n-k+1) =A001105(n), 1 <= k <= n.
T(n, 1) =A002061(n), central polygonal numbers.
Sum_{k=1..n} T(n, k) =A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) =A000537(n) (sum of first n rows).
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EXAMPLE
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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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n^2-n+2*k-1;
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MATHEMATICA
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Table[n^2-n+2*k-1, {n, 15}, {k, n}]//Flatten (*G. C. Greubel,Mar 10 2024 *)
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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
(Magma) [n^2-n+2*k-1: k in [1..n], n in [1..15]]; //G. C. Greubel,Mar 10 2024
(SageMath) flatten([[n^2-n+2*k-1 for k in range(1, n+1)] for n in range(1, 16)]) #G. C. Greubel,Mar 10 2024
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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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