OFFSET
1,1
LINKS
G. C. Greubel, Rows n = 1..100 of triangle, flattened
FORMULA
T(n, m) = ((m+f(m))*(m+2 - f(m))^(n+2) - (m-f(m))*(m+2 + f(m))^(n+2))/( 2^(n+3)*f(m)), where f(m) = sqrt(m*(m+4)).
From G. C. Greubel, Oct 08 2019: (Start)
T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2;
T(n, k) = (k*Fibonacci(n+2, m+2, -1) + Lucas(n+2, m+2, -1))/2, where Fibonacci(n, x, y) and Lucas(n, x, y) are the bi-variate Fibonacci an Lucas polynomials, respectively. (End)
EXAMPLE
Triangle begins as:
5;
13, 41;
34, 153, 436;
89, 571, 2089, 5741;
233, 2131, 10009, 33461, 90481;
MAPLE
seq(seq(simplify(( k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2) )/2), k=1..n), n=1..10); # G. C. Greubel, Oct 09 2019
MATHEMATICA
f[k_]:= Sqrt[k*(k+4)]; T[n_, m_]:= T[n, m]= FullSimplify[((m+f[m])*(m+2 - f[m])^(n+2) - (m-f[m])*(m+2 + f[m])^(n+2))/(2^(n+3)*f[m])]; Table[T[n, m], {n, 10}, {m, n}]//Flatten (* modified by G. C. Greubel, Oct 08 2019 *)
T[n_, k_]:= T[n, k]= (k*ChebyshevU[n, (k+2)/2] + 2*ChebyshevT[n+1, (k+ 2)/2])/2; Table[T[n, k], {n, 10}, {k, n}]/Flatten (* G. C. Greubel, Oct 08 2019 *)
PROG
(PARI) T(n, k)= ( k*sin((n+1)*acos((k+2)/2))/sin(acos((k+2)/2)) + 2*cos((n+1)*acos((k+2)/2)) )/2;
for(n=1, 10, for(k=1, n, print1(round(T(n, k)), ", "))) \\ G. C. Greubel, Oct 08 2019
(Magma)
T:= func< n, k | ( k*Sinh((n+1)*Argcosh((k+2)/2))/Sinh(Argcosh((k+2)/2)) + 2*Cosh((n+1)*Argcosh((k+2)/2)) )/2 >;
[Round(T(n, k)): k in [1..n], n in [1..10]]; // G. C. Greubel, Oct 08 2019
(Sage)
[[( k*chebyshev_U(n, (k+2)/2) + 2*chebyshev_T(n+1, (k+2)/2) )/2 for k in (1..n)] for n in (1..10)] # G. C. Greubel, Oct 08 2019
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 09 2006
EXTENSIONS
Major edit and new name, G. C. Greubel, Oct 08 2019
STATUS
approved