A144458 Two sequence determinant triangle sequence: a(n)=A000045(n); b(n)=b(n-1)+b(n-2)+b(n-3) :2 start;A141036(n); t(n,m)=t(n,m)=a(m)*b(n)-b(m)*a(n).

-2, -2, 0, -4, 2, 2, -6, 3, 3, 0, -10, 6, 6, 2, 3, -16, 13, 13, 10, 15, 17, -26, 25, 25, 24, 36, 47, 31, -42, 49, 49, 56, 84, 119, 119, 112, -68, 95, 95, 122, 183, 271, 318, 385, 329, -110, 182, 182, 254, 381, 580, 741, 991, 1127, 963

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 07 2008

Keywords

• uned
• sign

Comments

Row sums are:{-2, -2, 0, 0, 7, 52, 162, 546, 1730, 5291}.

Reasoning behind the sequence is:

Suppose we have n affine transforms that form a group:

g={ a(1)*x+b(1),a(2)*x+b(2),...,a(n)*x+b(n)}

on the sequences a(n) and b(n).

We form rational projections as Moebius / bilinear transforms:

g(projection)={( a(1)*x+b(1))/(a(n)*x+b(n)),( a(2)*x+b(2))/(a(n)*x+b(n)),...,( a(n-1)*x+b(n-1))/(a(n)*x+b(n))

With determinants:

g_det={a(1)*b(n)-b(1)*a(n),a(2)*b(n)-b(2)*a(n),...,a(n-1)*b(n)-b(n-1)*a(n)}

So that we have the triangular sequences:

t(n,m)=a(m)*b(n)-b(m)*a(n)

Examples

			{-2},
{-2, 0},
{-4, 2, 2},
{-6, 3, 3, 0},
{-10, 6, 6, 2, 3},
{-16, 13, 13, 10, 15, 17},
{-26, 25, 25, 24, 36, 47, 31},
{-42, 49, 49, 56, 84, 119, 119, 112},
{-68, 95, 95, 122, 183, 271, 318, 385, 329},
{-110, 182, 182, 254, 381, 580, 741, 991, 1127, 963}
		

Crossrefs

Cf. A141036, A000045.

Programs

Mathematica

Clear[a, b, t, n, m] a[n_] := Fibonacci[n]; b[0] = 2; b[1] = 1; b[2] = 1; b[n_] := b[n] = b[n - 1] + b[n - 2] + b[n - 3]; t[n_, m_] := a[m]*b[n] - b[m]*a[n]; Table[Table[t[n, m], {m, 0, n - 1}], {n, 1, 10}]; Flatten[%]

Formula

a(n)=A000045(n); b(n)=b(n-1)+b(n-2)+b(n-3) :2 start;A141036; t(n,m)=t(n,m)=a(m)*b(n)-b(m)*a(n).