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 A176271 The odd numbers as a triangle read by rows. 24
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified August 10 05:56 EDT 2024. Contains 375044 sequences. (Running on oeis4.)