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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

T(n,k) = A005408(n*(n-1)/2 + k - 1);

Nicomachus: row sums give A000578;

A000537(n) = sum of first n rows;

ABS(alternating row sums) give A065599;

central terms give A016754: T(2*n-1,n) = A016754(n-1);

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);

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

LINKS

Table of n, a(n) for n=1..66.

Eric Weisstein's World of Mathematics, Nicomachus's Theorem

Wikipedia, Nikomachos von Gerasa

FORMULA

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

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

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

CROSSREFS

Cf. 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 April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)