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A094727
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Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.
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22
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1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23
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OFFSET
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1,2
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COMMENTS
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All numbers m occur ceiling(m/2) times, see A004526.
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LINKS
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FORMULA
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T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009
a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)
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EXAMPLE
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Triangle begins:
1;
2, 3;
3, 4, 5;
4, 5, 6, 7;
5, 6, 7, 8, 9;
6, 7, 8, 9, 10, 11;
7, 8, 9, 10, 11, 12, 13;
8, 9, 10, 11, 12, 13, 14, 15;
9, 10, 11, 12, 13, 14, 15, 16, 17;
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MATHEMATICA
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PROG
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(Magma) z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];
(Haskell)
a094727 n k = n + k
a094727_row n = a094727_tabl !! (n-1)
a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
(SageMath) flatten([[n+k for k in range(n)] 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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