A255978 a(n) = a(n-1) + a(n-2) + (1 + (-1)^(a(n-1) + a(n-2))) with a(0)=0, a(1)=1.

0, 1, 1, 4, 5, 9, 16, 25, 41, 68, 109, 177, 288, 465, 753, 1220, 1973, 3193, 5168, 8361, 13529, 21892, 35421, 57313, 92736, 150049, 242785, 392836, 635621, 1028457, 1664080, 2692537, 4356617, 7049156, 11405773, 18454929, 29860704, 48315633, 78176337, 126491972, 204668309, 331160281, 535828592

Offset

0,4

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

G.f.: x*(1+2*x^2-x^3)/((1-x)*(1+x+x^2)*(1-x-x^2)). - Joerg Arndt, Mar 16 2015

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

a(n) = 2*Fibonacci(n) - (1 if n != 0 (mod 3)). - Nicolas Bělohoubek, Sep 29 2021

Example

For n = 2, a(2) = 0 + 1 + (1 + (-1)^(0 + 1)) = 1.
For n = 3, a(3) = 1 + 1 + (1 + (-1)^(1 + 1)) = 4.
For n = 4, a(4) = 1 + 4 + (1 + (-1)^(1 + 4)) = 5.
For n = 5, a(5) = 4 + 5 + (1 + (-1)^(4 + 5)) = 9.

Mathematica

RecurrenceTable[{a[n]==a[n-1]+a[n-2] +(1+(-1)^(a[n-1]+a[n-2])), a[0]==0, a[1]==1}, a, {n, 0, 50}]
CoefficientList[Series[x (1 + 2 x^2 - x^3) / ((1 - x) (1 + x + x^2) (1 - x - x^2)), {x, 0, 70}], x] (* Vincenzo Librandi, Mar 24 2015 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {0, 1, 1, 4, 5}, 50] (* Harvey P. Dale, Mar 26 2019 *)

Prog

(PARI) concat(0, Vec(x*(1+2*x^2-x^3)/((1-x)*(1+x+x^2)*(1-x-x^2)) + O(x^30))) \\ Michel Marcus, Mar 23 2015
(Magma) [n le 2 select (n-1) else Self(n-1)+Self(n-2)+(1+(-1)^(Self(n-1)+Self(n-2))): n in [1..45] ]; // Vincenzo Librandi, Mar 24 2015

Crossrefs

Cf. A000045, A253197, A253198.

Sequence in context: A155565 A278336 A051216 * A069089 A143096 A280225

Adjacent sequences: A255975 A255976 A255977 * A255979 A255980 A255981

Keyword

nonn,easy

Author

Waldemar Puszkarz, Mar 12 2015

Status

approved