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A253198 a(n) = a(n-1) + a(n-2) - (-1)^(a(n-1) + a(n-2)) with a(0)=0, a(1)=1. 3
0, 1, 2, 4, 5, 10, 16, 25, 42, 68, 109, 178, 288, 465, 754, 1220, 1973, 3194, 5168, 8361, 13530, 21892, 35421, 57314, 92736, 150049, 242786, 392836, 635621, 1028458, 1664080, 2692537, 4356618, 7049156, 11405773, 18454930, 29860704, 48315633, 78176338, 126491972, 204668309, 331160282, 535828592 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This is a minimally modified Fibonacci sequence (A000045) in that it preserves characteristic properties of the original sequence: a(n) is a function of the sum of the preceding two terms, the ratio of two consecutive terms tends to the Golden Mean, and the initial two terms are the same as in the Fibonacci sequence. See A253197 and A255978 for other members of this family.
LINKS
W. Puszkarz, A Note on Minimal Extensions of the Fibonacci Sequence, viXra:1503.0113, 2015.
FORMULA
a(n) = a(n-1) + a(n-2) - (-1)^(a(n-1) + a(n-2)), a(0)=0, a(1)=1.
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n>4. - Colin Barker, Mar 28 2015
G.f.: -x*(2*x^3-x^2-x-1) / ((x-1)*(x^2+x-1)*(x^2+x+1)). - Colin Barker, Mar 28 2015
a(n) = 2*A000045(n) - A079978(n+2). - Nicolas Bělohoubek, Aug 16 2021
EXAMPLE
For n=2, a(2) = 0 + 1 - (-1)^1 = 0 + 1 + 1 = 2.
For n=3, a(3) = 1 + 2 - (-1)^3 = 1 + 2 + 1 = 4.
For n=4, a(4) = 2 + 4 - (-1)^6 = 2 + 4 - 1 = 5.
MATHEMATICA
RecurrenceTable[{a[n]==a[n-1]+a[n-2] -(-1)^(a[n-1]+a[n-2]), a[0]==0, a[1]==1}, a, {n, 0, 50}]
LinearRecurrence[{1, 1, 1, -1, -1}, {0, 1, 2, 4, 5}, 50] (* Harvey P. Dale, Mar 17 2019 *)
PROG
(Magma) [n le 2 select (n-1) else Self(n-1) + Self(n-2) - (-1)^(Self(n-1) + Self(n-2)): n in [1..50] ]; // Vincenzo Librandi, Mar 28 2015
(PARI) concat(0, Vec(-x*(2*x^3-x^2-x-1)/((x-1)*(x^2+x-1)*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Mar 28 2015
CROSSREFS
Sequence in context: A276281 A059994 A299492 * A138856 A333188 A018401
KEYWORD
nonn,easy
AUTHOR
Waldemar Puszkarz, Mar 24 2015
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)