A176271 The odd numbers as a triangle read by rows.

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

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

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)

Maple

A176271 := proc(n, k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

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

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.

Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.

Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

Sequence in context: A317439 A004273 A005408 * A144396 A060747 A089684

Adjacent sequences: A176268 A176269 A176270 * A176272 A176273 A176274

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Apr 13 2010

Status

approved