cp's OEIS Frontend

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).

%I A102662 #23 Aug 19 2025 08:37:31
%S A102662 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,
%T A102662 1,15,83,211,255,143,35,3,1,17,111,353,579,489,207,41,3,1,19,143,547,
%U A102662 1143,1323,839,283,47,3,1,21,179,801,2043,3045,2651,1329,371,53,3,1,23,219
%N 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).
%C A102662 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.
%D A102662 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%H A102662 Reinhard Zumkeller, <a href="/A102662/b102662.txt">Rows n=0..149 of triangle, flattened</a>
%H A102662 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.
%F A102662 From _Clark Kimberling_, Feb 20 2012: (Start)
%F A102662 A102662=v and A207624=u, defined together as follows:
%F A102662 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,
%F A102662 where u(1,x)=1, v(1,x)=1; see the Mathematica section. (End)
%e A102662 Triangle begins:
%e A102662   1
%e A102662   1 3
%e A102662   1 5 3
%e A102662   1 7 11 3
%e A102662   1 9 23 17 3
%t A102662 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A102662 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A102662 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1
%t A102662 Table[Factor[u[n, x]], {n, 1, z}]
%t A102662 Table[Factor[v[n, x]], {n, 1, z}]
%t A102662 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A102662 TableForm[cu]
%t A102662 Flatten[%]    (* A207624 *)
%t A102662 Table[Expand[v[n, x]], {n, 1, z}]
%t A102662 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A102662 TableForm[cv]
%t A102662 Flatten[%]    (* A102662 *)
%t A102662 (* _Clark Kimberling_, Feb 20 2012 *)
%o A102662 (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))))))
%o A102662 BM(n) = M=matrix(n,n);for(i=1,n, for(j=1,n,M[i,j]=T(i,j)));M
%o A102662 M=BM(10)
%o A102662 for(i=1,10,s=0;for(j=1,i,s+=M[i,j]);print1(s,","))
%o A102662 (Haskell)
%o A102662 a102662 n k = a102662_tabl !! n !! k
%o A102662 a102662_row n = a102662_tabl !! n
%o A102662 a102662_tabl = [1] : [1,3] : f [1] [1,3] where
%o A102662    f xs ys = zs : f ys zs where
%o A102662      zs = zipWith (+) ([0] ++ xs ++ [0]) $
%o A102662                       zipWith (+) ([0] ++ ys) (ys ++ [0])
%o A102662 -- _Reinhard Zumkeller_, Feb 23 2012
%Y A102662 Cf. A038763, A048654, A008288.
%K A102662 nonn,tabl,changed
%O A102662 1,3
%A A102662 Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Feb 03 2005