A253198 a(n) = a(n-1) + a(n-2) - (-1)^(a(n-1) + a(n-2)) with a(0)=0, a(1)=1.

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

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

Colin Barker, Table of n, a(n) for n = 0..1000

W. Puszkarz, A Note on Minimal Extensions of the Fibonacci Sequence, viXra:1503.0113, 2015.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

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

Cf. A000045, A253197, A255978, A079978.

Sequence in context: A276281 A059994 A299492 * A138856 A333188 A018401

Adjacent sequences: A253195 A253196 A253197 * A253199 A253200 A253201

Keyword

nonn,easy

Author

Waldemar Puszkarz, Mar 24 2015

Status

approved