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A245323 a(n) = F(6*n-3)*(L(2*n-1)+1), where F = A000045 are the Fibonacci and L = A000032 are the Lucas numbers. 0
4, 170, 7320, 328380, 15124186, 704915600, 33014404692, 1549142827050, 72743819556328, 3416820019114700, 160507201018772634, 7540231471940495520, 354226959651753624100, 16641065639596669234730, 781774759322033582085816, 36726752905662141638238300 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..16.

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

Cf. A000045, A000032, A000042.

Sequence in context: A221083 A181191 A057140 * A195631 A145245 A081783

Adjacent sequences:  A245320 A245321 A245322 * A245324 A245325 A245326

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

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.