OFFSET
1,1
COMMENTS
Let n = 2*m-1 where m is 1,2,3,...; then a(m) = F(3*n)*(L(n)+1) with F(n) a Fibonacci number and L(n) a Lucas number (A000032); also a(n) = F(n)*(L(n)^3+L(n)^2+L(n)+1) which is a repdigit in base L(n), made of four digits F(n).
For n = 1, unary representation must be used to give a repdigit, then 4(10) = 1111(1).
For n = 3, 170(10) = 2222(4).
For n = 5, 7320(10) = 5555(11).
Starting from n = 5, the repdigit is the 4th term of a Fibonacci-type sequence of 5 palindromes.
For example this sequence for n=5 in base 11 is: 1331, 2112, 3443, 5555, 8998 which are the 5 2-digit Fibonacci numbers in base 10 concatenated with their reversed forms.
If the repdigit F(3*n)*(L(n)+1) is the most noticeable result in base L(n), there are other recurrences; naming f the digit F(n) and g the digit F(n)*2, we find
F(0*n)*(L(n)+1) = 0
F(1*n)*(L(n)+1) = ff
F(2*n)*(L(n)+1) = ff0
F(3*n)*(L(n)+1) = ffff
F(4*n)*(L(n)+1) = ffgg0
The base L(n) gives a visual aspect to the formula
F(n*k) = L(n)*F(n*(k-1)) + F(n*(k-2)) with n odd, which is a particular case of the general formula for any integers n,k,m:
F(n*(k)+m) = L(n) * F(n*(k-1)+m) - (-1)^n * F(n*(k-2)+m)
LINKS
Index entries for linear recurrences with constant coefficients, signature (72,-1304,6066,-1304,72,-1).
FORMULA
a((n-1)/2) = F(3*n)*(L(n)+1) for any positive odd n. [Corrected by M. F. Hasler, Oct 20 2016]
a(n) = F(n)*(L(n)^3+L(n)^2+L(n)+1).
G.f.: 2*x*(51*x^4-622*x^3+148*x^2-59*x+2) / ((x^2-47*x+1)*(x^2-18*x+1)*(x^2-7*x+1)). - Colin Barker, Jul 18 2014
EXAMPLE
Example: for n = 5, F(15) = 610, L(5) = 11, then a(5) = 610*12 = 7320 which is 5555 in base 11; F(5) = 5.
MATHEMATICA
LinearRecurrence[{72, -1304, 6066, -1304, 72, -1}, {4, 170, 7320, 328380, 15124186, 704915600}, 30] (* Harvey P. Dale, Aug 26 2014 *)
Table[Fibonacci[6 n - 3] (LucasL[2 n - 1] + 1), {n, 16}] (* Michael De Vlieger, Oct 21 2016 *)
PROG
(PARI) vector(50, m, fibonacci(6*m-3)*(lucas(2*m-1)+1)) \\ Colin Barker, Jul 18 2014
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémi Schulz, Jul 18 2014
EXTENSIONS
More terms from Colin Barker, Jul 18 2014
Edited and partially corrected by M. F. Hasler, Oct 09 and Oct 20 2016
STATUS
approved