OFFSET
1,5
COMMENTS
Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).
A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
1, 2, 3;
1, 3, 6, 4;
1, 4, 11, 14, 7;
1, 5, 18, 36, 34, 11;
1, 6, 27, 76, 119, 82, 18;
1, 7, 38, 140, 322, 393, 198, 29;
...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.
MAPLE
Lucas := proc(n, x) # see A114525
option remember;
if n=0 then
2;
elif n =1 then
x ;
else
x*procname(n-1, x)+procname(n-2, x) ;
end if;
expand(%) ;
end proc:
A117938 := proc(n::integer, k::integer)
if k = 1 then
1;
else
subs(x=n-k+1, Lucas(k-1, x)) ;
end if;
end proc:
seq(seq(A117938(n, k), k=1..n), n=1..12) ; # R. J. Mathar, Aug 16 2019
MATHEMATICA
T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)
PROG
(Sage)
def A117938(n, k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Apr 03 2006
EXTENSIONS
Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021
STATUS
approved