OFFSET
1,2
COMMENTS
A108309(n) = number of primes in n-th row.
LINKS
G. C. Greubel,Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics,Nicomachus's Theorem
Wikipedia,Nikomachos von Gerasa
FORMULA
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(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).
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).
EXAMPLE
FromPhilippe 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)
MAPLE
MATHEMATICA
Table[n^2-n+2*k-1, {n, 15}, {k, n}]//Flatten (*G. C. Greubel,Mar 10 2024 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller,Apr 13 2010
STATUS
approved