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 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). 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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 * A142048 A117563 A060439 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

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)