A253197 -id:A253197 - cp's OEIS frontend

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

Waldemar Puszkarz, Mar 12 2015

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.

			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.

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).

Cf. A000045, A253197, A253198.

[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

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 *)

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

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

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

Original entry on oeis.org

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

Waldemar Puszkarz, Mar 24 2015

			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.

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).

Cf. A000045, A253197, A255978, A079978.

[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

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 *)

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

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

A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 12, 16, 20, 24, 29, 34, 41, 44, 50, 57, 66, 74, 83, 91, 102, 112, 124, 135, 148, 161, 174, 187, 201, 214, 230, 246, 263, 279, 296, 313, 331, 349, 369, 388, 408, 428, 450, 471, 494, 516, 540, 563, 588, 612, 637, 662, 689, 715, 741, 769

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A253197, A126256.

import Data.Set (singleton, insert, size)
a258318 n = a258318_list !! n
a258318_list = f 2 a258197_tabl $ singleton 0 where
   f k (xs:xss) zs = g (take (div k 2) xs) zs where
     g []     ys = size ys : f (k + 1) xss ys
     g (x:xs) ys = g xs (insert x ys)

cp's OEIS Frontend

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

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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.

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A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

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