A247698 Brady numbers: B(n) = B(n - 1) + B(n - 2) with B(1) = 2308 and B(2) = 4261.

2308, 4261, 6569, 10830, 17399, 28229, 45628, 73857, 119485, 193342, 312827, 506169, 818996, 1325165, 2144161, 3469326, 5613487, 9082813, 14696300, 23779113, 38475413, 62254526, 100729939, 162984465, 263714404, 426698869, 690413273, 1117112142, 1807525415, 2924637557, 4732162972, 7656800529

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

Sequence in context: A031774 A031546 A250874 * A247839 A280659 A060231

Adjacent sequences: A247695 A247696 A247697 * A247699 A247700 A247701

Keyword

nonn,easy

Author

Sebastian Zimmer, Sep 22 2014

Status

approved