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

%I A051162 #89 Apr 24 2024 12:57:22
%S A051162 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,
%T A051162 11,12,13,14,8,9,10,11,12,13,14,15,16,9,10,11,12,13,14,15,16,17,18,10,
%U A051162 11,12,13,14,15,16,17,18,19,20
%N A051162 Triangle T(n,k) = n+k, n >= 0, 0 <= k <= n.
%C A051162 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
%C A051162 In partitions of n into distinct parts having maximal size, a(n) is the greatest number, see A000009. - _Reinhard Zumkeller_, Jun 13 2009
%C A051162 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
%C A051162 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
%C A051162 If we subtract 1 from every term the result is essentially A213183. - _N. J. A. Sloane_, Apr 28 2020
%H A051162 Reinhard Zumkeller, <a href="/A051162/b051162.txt">Rows n=0..100 of triangle, flattened</a>
%H A051162 Eric Naslund, <a href="http://math.stackexchange.com/questions/46713/euler-mascheroni-constant-expression-further-simplification/46718#46718">Euler-Mascheroni constant expression, further simplification</a>
%H A051162 Dmitry A. Zaitsev, <a href="https://doi.org/10.1016/j.tcs.2016.11.002">A generalized neighborhood for cellular automata</a>, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.
%F A051162 T(n, k) = n + k, 0 <= k <= n.
%F A051162 a(n-1) = 2*A002260(n) + A004736(n) - 3, n > 0. - _Boris Putievskiy_, Mar 12 2012
%F A051162 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
%F A051162 From _Robert Israel_, May 21 2015: (Start)
%F A051162 a(n) = A003056(n) + A002262(n).
%F A051162 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)
%F A051162 G.f. as triangle: (x + (2 - 3*x)*x*y)/((1 - x)^2*(1 - x*y)^2). - _Stefano Spezia_, Apr 22 2024
%e A051162 The triangle  T(n, k) starts:
%e A051162 n\k  0  1  2  3  4  5  6  7  8  9 10 ...
%e A051162 0:   0
%e A051162 1:   1  2
%e A051162 2:   2  3  4
%e A051162 3:   3  4  5  6
%e A051162 4:   4  5  6  7  8
%e A051162 5:   5  6  7  8  9 10
%e A051162 6:   6  7  8  9 10 11 12
%e A051162 7:   7  8  9 10 11 12 13 14
%e A051162 8:   8  9 10 11 12 13 14 15 16
%e A051162 9:   9 10 11 12 13 14 15 16 17 18
%e A051162 10: 10 11 12 13 14 15 16 17 18 19 20
%e A051162 ... reformatted. - _Wolfdieter Lang_, May 15 2015
%p A051162 seq(seq(r+c, c=0..r),r=0..10); # _Robert Israel_, May 21 2015
%t A051162 With[{c=Range[0,20]}, Flatten[Table[Take[c,{n,2n-1}], {n,11}]]] (* _Harvey P. Dale_, Nov 19 2011 *)
%o A051162 (Haskell)
%o A051162 a051162 n k = a051162_tabl !! n !! k
%o A051162 a051162_row n = a051162_tabl !! n
%o A051162 a051162_tabl = iterate (\xs@(x:_) -> (x + 1) : map (+ 2) xs) [0]
%o A051162 -- _Reinhard Zumkeller_, Sep 17 2014, Oct 02 2012, Apr 23 2012
%o A051162 (PARI) for(n=0,10,for(k=0,n,print1(n+k,", "))) \\ _Derek Orr_, May 19 2015
%Y A051162 Cf. A000009, A000217, A002260, A025581, A004736, A045943, A105125, A213183.
%Y A051162 Cf. also A008585 (central terms), A005843 (right edge).
%Y A051162 Cf. also A002262, A001477, A003056.
%K A051162 nonn,tabl,easy,nice,look
%O A051162 0,3
%A A051162 _N. J. A. Sloane_