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A245323
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a(n) = F(6*n-3)*(L(2*n-1)+1), where F = A000045 are the Fibonacci and L = A000032 are the Lucas numbers.
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0
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4, 170, 7320, 328380, 15124186, 704915600, 33014404692, 1549142827050, 72743819556328, 3416820019114700, 160507201018772634, 7540231471940495520, 354226959651753624100, 16641065639596669234730, 781774759322033582085816, 36726752905662141638238300
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) vector(50, m, fibonacci(6*m-3)*(lucas(2*m-1)+1)) \\ Colin Barker, Jul 18 2014
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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Edited and partially corrected by M. F. Hasler, Oct 09 and Oct 20 2016
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STATUS
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approved
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