A102662 - OEIS

A102662

Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

1, 1, 3, 1, 5, 3, 1, 7, 11, 3, 1, 9, 23, 17, 3, 1, 11, 39, 51, 23, 3, 1, 13, 59, 113, 91, 29, 3, 1, 15, 83, 211, 255, 143, 35, 3, 1, 17, 111, 353, 579, 489, 207, 41, 3, 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3, 1, 21, 179, 801, 2043, 3045, 2651, 1329, 371, 53, 3, 1, 23, 219

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OFFSET

1,3

COMMENTS

Generalization of A008288 (use initial terms 1,1,3). Triangle seen as lower triangular matrix: The absolute values of the coefficients of the characteristic polynomials of the n X n matrix are the (n+1)th row of A038763. Row sums give A048654.

REFERENCES

Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids" Fibonacci Association, 1993, p. 37

LINKS

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened

FORMULA

A102662=v and A207624=u, defined together as follows:

u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1; see the Mathematica section.

[From Clark Kimberling, Feb 20 2012]

EXAMPLE

Triangle begins:

1 3

1 5 3

1 7 11 3

1 9 23 17 3

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + v[n - 1, x]

v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1

Table[Factor[u[n, x]], {n, 1, z}]

Table[Factor[v[n, x]], {n, 1, z}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%] (* A207624 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%] (* A102662 *)

(* Clark Kimberling, Feb 20 2012 *)

PROG

(PARI) T(k, r)=if(r>k, 0, if(k==1, 1, if(k==2, if(r==1, 1, 3), if(r==1, 1, if(r==k, 3, T(k-1, r-1)+T(k-1, r)+T(k-2, r-1)))))) BM(n) = M=matrix(n, n); for(i=1, n, for(j=1, n, M[i, j]=T(i, j))); M M=BM(10) for(i=1, 10, s=0; for(j=1, i, s+=M[i, j]); print1(s, ", "))

(Haskell)

a102662 n k = a102662_tabl !! n !! k

a102662_row n = a102662_tabl !! n

a102662_tabl = [1] : [1, 3] : f [1] [1, 3] where

f xs ys = zs : f ys zs where

zs = zipWith (+) ([0] ++ xs ++ [0]) $

zipWith (+) ([0] ++ ys) (ys ++ [0])

-- Reinhard Zumkeller, Feb 23 2012

CROSSREFS

Cf. A038763, A048654, A008288.

Sequence in context: A208607 A159291 A122510 * A350279 A142048 A117563

Adjacent sequences: A102659 A102660 A102661 * A102663 A102664 A102665

KEYWORD

nonn,tabl

AUTHOR

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 03 2005

STATUS

approved