A004699 -id:A004699 - cp's OEIS frontend

A286311 a(n) = 2*a(n-1) - a(n-2) + a(n-4), n>3, a(0)=0, a(1)=a(2)=1, a(3)=3.

0, 1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, 349, 565, 915, 1481, 2396, 3876, 6271, 10147, 16419, 26567, 42986, 69552, 112537, 182089, 294627, 476717, 771344, 1248060, 2019403, 3267463, 5286867, 8554331, 13841198, 22395528, 36236725, 58632253, 94868979

Offset: 0

Paul Curtz, May 06 2017

Difference table for a(n):
   0,  1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, ...
   1,  0, 2, 2, 3, 4,  7, 12, 20, 32, 51,  82, 133, ...
  -1,  2, 0, 1, 1, 3,  5,  8, 12, 19, 31,  51,  83, ...
   3, -2, 1, 0, 2, 2,  3,  4,  7, 12, 20,  32,  51, ...
etc.
The pair a(n) = 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, ...
     and b(n) = 0, 2, 2, 3, 4, 7, 12, 20, 32, 51, ...
is interesting. a(n) and b(n) are autosequences of the first kind (see Link). a(n) and b(n) have the same first trisection: 3*A001076(n).
a(n) + b(n) = A022086(n) = 3*A000045(n) (Fibonacci).
b(n) - a(n) = 0, 1, 1, 0, -1, -1, 0, ... = A128834(n).
a(n+6) - a(n) = b(n+6) - b(n) = 6*Fib(n+3).
a(n) - a(n) mod 9 = 9*A004699(n) = b(n) - b(n) mod 9.

Colin Barker, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A000034, A000045, A001076, A004699, A022086, A128834.

I:=[0,1,1,3]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{2, -1, 0, 1}, {0, 1, 1, 3}, 40] (* or *)
CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x + x^2) (1 - x - x^2)), {x, 0, 39}], x] (* Michael De Vlieger, May 07 2017 *)

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)) + O(x^60))) \\ Colin Barker, May 06 2017

a(n) = 2*a(n-1) - a(n-2) + a(n-4). Valid for b(n).

G.f.: x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)). - Colin Barker, May 06 2017

More terms from Colin Barker, May 06 2017

A214286 a(n) = floor(Fibonacci(n)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 7, 12, 20, 33, 53, 87, 141, 228, 369, 597, 966, 1563, 2530, 4093, 6624, 10717, 17341, 28059, 45401, 73461, 118862, 192324, 311187, 503511, 814698, 1318209, 2132907, 3451116, 5584024, 9035140, 14619165

Offset: 0

Vincenzo Librandi, Jul 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1).

Cf. A004695-A004699.

Magma
```
[Floor(Fibonacci(n)/7): n in [0..40]];
```

Floor[Fibonacci[Range[0, 40]]/7] (* modified by G. C. Greubel, May 22 2019 *)
LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1},{0,0,0,0,0,0,1,1,3,4,7,12,20,33,53,87,141,228},50] (* Harvey P. Dale, Dec 01 2020 *)

vector(40, n, n--; fibonacci(n)\7 ) \\ G. C. Greubel, May 22 2019

[floor(fibonacci(n)/7) for n in (0..40)] # G. C. Greubel, May 22 2019

G.f.: x^6*(1+x^2+x^5+x^6+x^7+x^9+x^10) / ( (1-x-x^2)*(1-x^16) ). - R. J. Mathar, Jul 14 2012

a(n) = (A000045(n) - A105870(n))/7. - R. J. Mathar, Jul 14 2012

cp's OEIS Frontend

A286311 a(n) = 2*a(n-1) - a(n-2) + a(n-4), n>3, a(0)=0, a(1)=a(2)=1, a(3)=3.

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