OFFSET
1,1
COMMENTS
B(n) / B(n - 1) approaches the golden ratio as n approaches infinity.
LINKS
Logan Cooper, Table of n, a(n) for n = 1..1000 (truncated from 9966 to 1000 terms by M. F. Hasler, May 10 2017)
Brady Haran and Matt Parker, Brady Numbers, Numberphile video (2014).
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
a(n) = a(n-1) + a(n-2).
G.f.: x*(2308 + 1953*x) / (1-x-x^2). - Colin Barker, Sep 23 2014
a(n) = k*phi^n + o(1), where k = 976.5 + sqrt(354578.45) = 1571.96.... - Charles R Greathouse IV, Sep 28 2014
a(n) = 2308*A000045(n-2) + 4261*A000045(n-1) = 1953*A000045(n+1) + 355*A000045(n). - M. F. Hasler, May 10 2017
a(n) = F(n+17) - F(n+8) - 9*F(n) - F(n-14) for F(n) = A000045(n). - Greg Dresden, Jul 07 2022
MAPLE
Brady1 := proc(n::posint)
option remember, system;
if n = 1 then
2308
elif n = 2 then
4261
else
thisproc( n - 1 ) + thisproc( n - 2 )
end if
end proc:
seq( Brady1( n ), n = 1 .. 100 );
# James McCarron, Oct 05 2019
# alternate program
Brady2 := ( n :: posint ) -> coeff( series(x*(2308+1953*x)/(1-x-x^2), x, n+1), x^n ):
seq( Brady2( n ), n = 1 .. 100 );
# James McCarron, Oct 05 2019
MATHEMATICA
LinearRecurrence[{1, 1}, {2308, 4261}, n]
Rest[CoefficientList[Series[x*(2308+1953*x)/(1-x-x^2), {x, 0, 50}], x]] (* G. C. Greubel, Sep 07 2018 *)
PROG
(Haskell) brady = let makeSeq a b = a : makeSeq b (a + b) in makeSeq 2308 4261
(PARI) Vec(-x*(1953*x+2308)/(x^2+x-1) + O(x^50)) \\ Colin Barker, Sep 23 2014
(PARI) a(n)=([1, 1; 1, 0]^n*[1953; 355])[1, 1] \\ Charles R Greathouse IV, Jan 20 2016
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2308+1953*x)/(1-x-x^2))); // G. C. Greubel, Sep 07 2018
(Python)
def A247698_list(n):
list = [2308, 4261] + [0] * (n - 2)
for i in range(2, n):
list[i] = list[i - 1] + list[i - 2]
return list
print(A247698_list(32)) # M. Eren Kesim, Jun 28 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sebastian Zimmer, Sep 22 2014
STATUS
approved