A051162 Triangle T(n,k) = n+k, n >= 0, 0 <= k <= n.

0, 1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice
• look

Comments

Row sums are A045943 = triangular matchstick numbers: 3n(n+1)/2. This was independently noted by me and, without cross-reference, as a comment on A045943, by Jon Perry, Jan 15 2004. - Jonathan Vos Post, Nov 09 2007

In partitions of n into distinct parts having maximal size, a(n) is the greatest number, see A000009. - Reinhard Zumkeller, Jun 13 2009

Row sums of reciprocals of terms in this triangle converge to log(2). See link to Eric Naslund's answer. - Mats Granvik, Mar 07 2013

T(n,k) satisfies the cubic equation T(n,k)^3 + 3*A025581(n, k)*T(n,k) - 4*A105125(n,k) = 0. This is a problem similar to the one posed by François Viète (Vieta) mentioned in a comment on A025581. Here the problem is to determine for a rectangle (a, b), with a > b >= 1, from the given values for a^3 + b^3 and a - b the value of a + b. Here for nonnegative integers a = n and b = k. - Wolfdieter Lang, May 15 2015

If we subtract 1 from every term the result is essentially A213183. - N. J. A. Sloane, Apr 28 2020

Examples

			The triangle  T(n, k) starts:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   0
1:   1  2
2:   2  3  4
3:   3  4  5  6
4:   4  5  6  7  8
5:   5  6  7  8  9 10
6:   6  7  8  9 10 11 12
7:   7  8  9 10 11 12 13 14
8:   8  9 10 11 12 13 14 15 16
9:   9 10 11 12 13 14 15 16 17 18
10: 10 11 12 13 14 15 16 17 18 19 20
... reformatted. - _Wolfdieter Lang_, May 15 2015
		

Links

• Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
• Eric Naslund, Euler-Mascheroni constant expression, further simplification
• Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.

Crossrefs

Cf. A000009, A000217, A002260, A025581, A004736, A045943, A105125, A213183.

Cf. also A008585 (central terms), A005843 (right edge).

Cf. also A002262, A001477, A003056.

Programs

Haskell

a051162 n k = a051162_tabl !! n !! k
a051162_row n = a051162_tabl !! n
a051162_tabl = iterate (\xs@(x:_) -> (x + 1) : map (+ 2) xs) [0]
-- Reinhard Zumkeller, Sep 17 2014, Oct 02 2012, Apr 23 2012

Maple

seq(seq(r+c, c=0..r),r=0..10); # Robert Israel, May 21 2015

Mathematica

With[{c=Range[0,20]}, Flatten[Table[Take[c,{n,2n-1}], {n,11}]]] (* Harvey P. Dale, Nov 19 2011 *)

PARI

for(n=0,10,for(k=0,n,print1(n+k,", "))) \\ Derek Orr, May 19 2015

Formula

T(n, k) = n + k, 0 <= k <= n.

a(n-1) = 2*A002260(n) + A004736(n) - 3, n > 0. - Boris Putievskiy, Mar 12 2012

a(n-1) = (t - t^2+ 2n-2)/2, where t = floor((-1+sqrt(8*n-7))/2), n > 0. - Robert G. Wilson v and Boris Putievskiy, Mar 14 2012

From Robert Israel, May 21 2015: (Start)

a(n) = A003056(n) + A002262(n).

G.f.: x/(1-x)^2 + (1-x)^(-1)*Sum(j>=1, (1-j)*x^A000217(j)). The sum is related to Jacobi Theta functions. (End)

G.f. as triangle: (x + (2 - 3*x)*x*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Apr 22 2024