A176271 - OEIS

%I #28 Mar 10 2024 09:33:44

%S 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,

%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131

%N The odd numbers as a triangle read by rows.

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

%H G. C. Greubel, <a href="/A176271/b176271.txt">Rows n = 1..100 of the triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NicomachussTheorem.html">Nicomachus's Theorem</a>

%H Wikipedia, <a href="http://de.wikipedia.org/wiki/Nikomachos_von_Gerasa">Nikomachos von Gerasa</a>

%F T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.

%F T(n, k) = A005408(n*(n-1)/2 + k - 1).

%F T(2*n-1, n) = A016754(n-1) (main diagonal).

%F T(2*n, n) = A000466(n).

%F T(2*n, n+1) = A053755(n).

%F T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.

%F T(n, 1)   = A002061(n), central polygonal numbers.

%F T(n, 2)   = A027688(n-1) for n > 1.

%F T(n, 3)   = A027690(n-1) for n > 2.

%F T(n, 4)   = A027692(n-1) for n > 3.

%F T(n, 5)   = A027694(n-1) for n > 4.

%F T(n, 6)   = A048058(n-1) for n > 5.

%F T(n, n-3) = A108195(n-2) for n > 3.

%F T(n, n-2) = A082111(n-2) for n > 2.

%F T(n, n-1) = A014209(n-1) for n > 1.

%F T(n, n) = A028387(n-1).

%F Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).

%F Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

%e From _Philippe Deléham_, Oct 03 2011: (Start)

%e Triangle begins:

%e    1;

%e    3,  5;

%e    7,  9, 11;

%e   13, 15, 17, 19;

%e   21, 23, 25, 27, 29;

%e   31, 33, 35, 37, 39, 41;

%e   43, 45, 47, 49, 51, 53, 55;

%e   57, 59, 61, 63, 65, 67, 69, 71;

%e   73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

%p A176271 := proc(n,k)

%p     n^2-n+2*k-1 ;

%p end proc: # _R. J. Mathar_, Jun 28 2013

%t Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 10 2024 *)

%o (Haskell)

%o a176271 n k = a176271_tabl !! (n-1) !! (k-1)

%o a176271_row n = a176271_tabl !! (n-1)

%o a176271_tabl = f 1 a005408_list where

%o    f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws

%o -- _Reinhard Zumkeller_, May 24 2012

%o (Magma) [n^2-n+2*k-1: k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 10 2024

%o (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

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

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

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

%K nonn,tabl

%O 1,2

%A _Reinhard Zumkeller_, Apr 13 2010