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

Author

Reinhard Zumkeller, Apr 13 2010

Keywords

• nonn
• tabl

Comments

A108309(n) = number of primes in n-th row.

Examples

			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)
		

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

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.

Programs

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

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 *)

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

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).