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URL: https://oeis.org/A117938

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A117938
Triangle, columns generated from Lucas Polynomials.
6
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, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123
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
FORMULA
Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...
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
(SageMath)
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
Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).
Sequence in context: A152976 A153861 A118981 * A256193 A101912 A208522
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Apr 03 2006
EXTENSIONS
Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021
STATUS
approved