

A106314


Triangle composed of squares, row sums = Paraffin numbers.


5



1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 25, 16, 9, 4, 1
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OFFSET

1,5


COMMENTS

Row sums = A005993, Paraffin numbers: 1, 2, 6, 10, 19, 28, 44, 60...
Row sums are; {1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146,...}


LINKS

Table of n, a(n) for n=1..55.


FORMULA

Given the triangle of A003983, replace each of the terms by its square.
p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= ( then than equal) Floor[n/2], 2*i + 1, (2*(n  i) + 1)]], {i, 0, n}]/(1  x);
t(n,m)=coefficients(p(x,n),x)


EXAMPLE

The triangle of A003983 is:
1;
1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 2, 3, 2, 1;
...
Replacing each term by its square, we get:
1;
1, 1;
1, 4, 1;
1, 4, 4, 1;
1, 4, 9, 4, 1;
...
{1},
{1, 1},
{1, 4, 1},
{1, 4, 4, 1},
{1, 4, 9, 4, 1},
{1, 4, 9, 9, 4, 1},
{1, 4, 9, 16, 9, 4, 1},
{1, 4, 9, 16, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1} (End)


MATHEMATICA

Clear[p, n, i];
p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*i + 1, (2*(n  i) + 1)]], {i, 0, n}]/(1  x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]


CROSSREFS

Cf. A003983, A106314, A005993.
Sequence in context: A046596 A174093 A204028 * A152716 A183374 A176263
Adjacent sequences: A106311 A106312 A106313 * A106315 A106316 A106317


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Apr 28 2005


EXTENSIONS

Additional comments from Roger L. Bagula and Gary W. Adamson, Apr 02 2009


STATUS

approved



