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A247698
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Brady numbers: B(n) = B(n - 1) + B(n - 2) with B(1) = 2308 and B(2) = 4261.
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3
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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
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OFFSET
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1,1
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COMMENTS
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B(n) / B(n - 1) approaches the golden ratio as n approaches infinity.
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LINKS
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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).
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A031774 A031546 A250874 * A247839 A280659 A060231
Adjacent sequences: A247695 A247696 A247697 * A247699 A247700 A247701
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KEYWORD
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nonn,easy
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AUTHOR
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Sebastian Zimmer, Sep 22 2014
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STATUS
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approved
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