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LucasL

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

LucasL[n]
    gives the nth Lucas number L_n.
LucasL[n, x]
    gives the nth Lucas polynomial L_n(x).
Exact integer orders are computed via GMP fast doubling (numbers, using
L_m = 2 F_{m+1} - F_m) or the recurrence L_k = x L_{k-1} + L_{k-2} with
L_0 = 2, L_1 = x (polynomials); negative orders use L_{-n} = (-1)^n L_n.
Inexact or complex orders evaluate the generalized closed form
L_n = phi^n + Cos[Pi n] phi^-n (phi = GoldenRatio) numerically.
Listable; symbolic orders stay unevaluated.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Table[LucasL[n], {n, 10}]
Out[1]= {1, 3, 4, 7, 11, 18, 29, 47, 76, 123}

In[2]:= LucasL[7, x]
Out[2]= 7 x + 14 x^3 + 7 x^5 + x^7

In[3]:= LucasL[-11.]
Out[3]= -199.0

In[4]:= N[LucasL[11/3], 50]
Out[4]= 5.92396265296195541013569786219401262875198554223617

Implementation notes

Algorithm. builtin_lucasl mirrors Fibonacci. For exact integer order it fast-doubles the Fibonacci pair (F_m, F_{m+1}) in GMP and derives L_m = 2 F_{m+1} - F_m in O(log n), with negative orders via L_{-m} = (-1)^m L_m. LucasL[n, x] (Lucas polynomial) iterates L_k = x L_{k-1} + L_{k-2} from L_0 = 2, L_1 = x, Expand-ing each step. For inexact/non-integer order it builds the closed form phi^n + Cos[Pi n] phi^-n (phi = GoldenRatio, or beta = (x+Sqrt[x^2+4])/2) and hands it to numericalize. Purely symbolic order returns NULL.

Data structures. GMP mpz_t integer pair; Expr trees through eval_and_free for the polynomial and closed-form branches.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= LucasL[10]
Out[1]= 123

In[1]:= LucasL[5, x]
Out[1]= 5 x + 5 x^3 + x^5

In[1]:= LucasL[100]
Out[1]= 792070839848372253127

In[1]:= LucasL[-7]
Out[1]= -29

In[1]:= N[LucasL[20]/LucasL[19], 20]
Out[1]= 1.6180339887498948482

Notes

LucasL[n] is the nth Lucas number L_n (L_0 = 2, L_1 = 1, L_k = L_{k-1} + L_{k-2}); LucasL[n, x] is the Lucas polynomial L_n(x). Integer orders use GMP fast doubling for arbitrary size (so LucasL[100] is exact), and negative orders follow L_{-n} = (-1)^n L_n. Consecutive ratios L_{n+1}/L_n converge to GoldenRatio (last example). Inexact or complex orders use the closed form phi^n + Cos[Pi n] phi^-n. Listable; symbolic orders stay unevaluated.