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A176353 A symmetrical triangle sequence based on Dirichlet's divisors:g(n)=n*Log[n] - n + Sqrt[n];t(n,m)=If[m == 0 || m == n, 1, 1 + Round[ -g(m) - g(n - m) + g(n)]] 0

%I

%S 1,1,1,1,2,1,1,2,2,1,1,3,3,3,1,1,3,3,3,3,1,1,3,4,4,4,3,1,1,3,4,5,5,4,

%T 3,1,1,3,4,5,5,5,4,3,1,1,3,5,6,6,6,6,5,3,1,1,3,5,6,6,7,6,6,5,3,1

%N A symmetrical triangle sequence based on Dirichlet's divisors:g(n)=n*Log[n] - n + Sqrt[n];t(n,m)=If[m == 0 || m == n, 1, 1 + Round[ -g(m) - g(n - m) + g(n)]]

%C Row sums are: {1, 2, 4, 6, 11, 14, 20, 26, 31, 42, 49,...}.

%C One possible method of relating sum form symmetrical triangles to product (factorial like) form triangles is that the sums forms are related to divisors.

%C The Dirichlet divisor approximate function for the factorial (here g(n)) gives a triangle at the exponential level that is here made into integers using the Round[] function.

%D George E. Andrews, Number Theory,Dover Publications,N.Y. 1971, pp 207-208

%F g(n)=n*Log[n] - n + Sqrt[n];

%F t(n,m)=If[m == 0 || m == n, 1, 1 + Round[ -g(m) - g(n - m) + g(n)]]

%e {1},

%e {1, 1},

%e {1, 2, 1},

%e {1, 2, 2, 1},

%e {1, 3, 3, 3, 1},

%e {1, 3, 3, 3, 3, 1},

%e {1, 3, 4, 4, 4, 3, 1},

%e {1, 3, 4, 5, 5, 4, 3, 1},

%e {1, 3, 4, 5, 5, 5, 4, 3, 1},

%e {1, 3, 5, 6, 6, 6, 6, 5, 3, 1},

%e {1, 3, 5, 6, 6, 7, 6, 6, 5, 3, 1}

%t g[n_] = n*Log[n] - n + Sqrt[n];

%t t1[n_, m_] = If[m == 0 || m == n, 1, 1 + Round[ -g[m] - g[n - m] + g[n]]];

%t Table[Table[t1[n, m], {m, 0, n}], {n, 0, 10}];

%t Flatten[%]

%Y Cf. A176346

%K nonn,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Apr 15 2010

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Last modified December 4 15:36 EST 2021. Contains 349526 sequences. (Running on oeis4.)