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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.

Wikipedia, Nikomachos von Gerasa

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

FORMULA

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

EXAMPLE

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 ;...[From Philippe Deléham, Oct 03 2011].

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: A247328 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 September 25 13:06 EDT 2017. Contains 292473 sequences.