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)]]

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, 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

Offset

0,5

Comments

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

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

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.

References

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

Links

Table of n, a(n) for n=0..65.

Formula

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)]]

Example

{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, 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}

Mathematica

g[n_] = n*Log[n] - n + Sqrt[n];
t1[n_, m_] = If[m == 0 || m == n, 1, 1 + Round[ -g[m] - g[n - m] + g[n]]];
Table[Table[t1[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A176346

Sequence in context: A106254 A117147 A111007 * A103691 A103441 A081206

Adjacent sequences: A176350 A176351 A176352 * A176354 A176355 A176356

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Apr 15 2010

Status

approved