A117647 -id:A117647 - cp's OEIS frontend

A173343 a(n+4) = a(n+3) - 2*a(n+2) - a(n+1) - a(n).

1, 2, 0, -5, -8, 0, 21, 34, 0, -89, -144, 0, 377, 610, 0, -1597, -2584, 0, 6765, 10946, 0, -28657, -46368, 0, 121393, 196418, 0, -514229, -832040, 0, 2178309, 3524578, 0, -9227465, -14930352, 0, 39088169, 63245986, 0, -165580141, -267914296, 0

Offset: 0

Creighton Dement, Feb 16 2010

Sequence appears to give signed Fibonacci numbers, where those Fibonacci numbers "missing" are in A173344. A117647 gives a nonnegative version without zeros. (a(n)) = kjbseq(X) with X = -0.25'i + 0.5'j + 0.5'k + 0.25'i + j' + 0.5k' - 0.25ii - 0.25'jj' - 0.25'kk' + 0.5'ij' + 0.5'ik' - 0.5'ji' -0.25'jk' + 0.25'kj' + 0.25'ee' (see Munafo link for definitions)

C. Dement, Online Floretion Multiplier [broken link]
R. J. Mathar, Structure of the Floretion Group
R. Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (1,-2,-1,-1).

Cf. A173344, A117647.

LinearRecurrence[{1,-2,-1,-1},{1,2,0,-5},50] (* Harvey P. Dale, Jul 17 2018 *)

G.f.: (x+1)/(x^4+x^3+2*x^2-x+1).

a(n) = b(n)+b(n-1) where b(3n) = b(3n+1) = -b(3n+2) = (-1)^n*A001076(n+1). [From R. J. Mathar, Apr 01 2010]

A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 8, 13, 20, 30, 21, 29, 42, 62, 92, 34, 55, 84, 126, 188, 280, 89, 123, 178, 262, 388, 576, 856, 144, 233, 356, 534, 796, 1184, 1760, 2616, 377, 521, 754, 1110, 1644, 2440, 3624, 5384, 8000, 610, 987, 1508, 2262, 3372, 5016, 7456, 11080, 16464, 24464

Offset: 1

Michel Lagneau, Jan 02 2019

Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1)), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) =  F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) =  F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) =  F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).

			The start of the sequence as a triangular array T(n, k) read by rows:
   1;
   2,   3;
   5,   7,  10;
   8,  13,  20,   30;
  21,  29,  42,   62,   92;
  34,  55,  84,  126,  188,  280;
  ...

Cf. A000045, A002878, A033887, A035312, A036561, A117647.

with(combinat,fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
  for j from 1 to 1000 while(ii=0) do:
     i:=fibonacci(j):
     if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
      then
      lst1:=lst1 union {i}:ii:=1:
      else
     fi:
   od:
    for k from 1 to n-1 do:
      lst1:=lst1 union {lst1[k]+lst[k]}:
    od:
    lst:=lst1:lst2:=lst2 union lst:
    print(lst1):
   od:

cp's OEIS Frontend

A173343 a(n+4) = a(n+3) - 2*a(n+2) - a(n+1) - a(n).

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