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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

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;

  ...

MAPLE

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:

CROSSREFS

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

Sequence in context: A265249 A214331 A182483 * A163975 A267521 A202267

Adjacent sequences:  A323010 A323011 A323012 * A323014 A323015 A323016

KEYWORD

nonn,tabl

AUTHOR

Michel Lagneau, Jan 02 2019

STATUS

approved

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Last modified April 25 04:45 EDT 2019. Contains 322451 sequences. (Running on oeis4.)