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 A051162 Triangle T(n,k) = n+k, n >= 0, 0 <= k <= n. 23
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums are A045943 = triangular matchstick numbers: 3n(n+1)/2. This was independently noted by myself 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 LINKS Reinhard Zumkeller, Rows n=0..100 of triangle, flattened Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35; https://doi.org/10.1016/j.tcs.2016.11.002 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) EXAMPLE 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 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 *) PROG (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 (PARI) for(n=0, 10, for(k=0, n, print1(n+k, ", "))) \\ Derek Orr, May 19 2015 CROSSREFS Cf. A025581, A004736, A045943, A213183. Cf. also A008585 (central terms), A005843 (right edge). Cf. also A002262, A001477, A003056. Sequence in context: A204006 A106251 A134478 * A122872 A132919 A162619 Adjacent sequences:  A051159 A051160 A051161 * A051163 A051164 A051165 KEYWORD nonn,tabl,easy,nice,look AUTHOR STATUS approved

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Last modified August 15 13:27 EDT 2020. Contains 336504 sequences. (Running on oeis4.)