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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.

%I A323013 #14 Mar 18 2025 07:23:50
%S A323013 1,2,3,5,7,10,8,13,20,30,21,29,42,62,92,34,55,84,126,188,280,89,123,
%T A323013 178,262,388,576,856,144,233,356,534,796,1184,1760,2616,377,521,754,
%U A323013 1110,1644,2440,3624,5384,8000,610,987,1508,2262,3372,5016,7456,11080,16464,24464
%N 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.
%C A323013 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, ...
%C A323013 We observe interesting properties:
%C A323013 T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
%C A323013 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, ...
%C A323013 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).
%C A323013 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).
%C A323013 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).
%C A323013 Other property:
%C A323013 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).
%C A323013 T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).
%e A323013 The start of the sequence as a triangular array T(n, k) read by rows:
%e A323013    1;
%e A323013    2,   3;
%e A323013    5,   7,  10;
%e A323013    8,  13,  20,   30;
%e A323013   21,  29,  42,   62,   92;
%e A323013   34,  55,  84,  126,  188,  280;
%e A323013   ...
%p A323013 with(combinat,fibonacci):
%p A323013 lst:={1}:lst2:=lst:
%p A323013 for n from 2 to 15 do :
%p A323013 lst1:={}:ii:=0:
%p A323013   for j from 1 to 1000 while(ii=0) do:
%p A323013      i:=fibonacci(j):
%p A323013      if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
%p A323013       then
%p A323013       lst1:=lst1 union {i}:ii:=1:
%p A323013       else
%p A323013      fi:
%p A323013    od:
%p A323013     for k from 1 to n-1 do:
%p A323013       lst1:=lst1 union {lst1[k]+lst[k]}:
%p A323013     od:
%p A323013     lst:=lst1:lst2:=lst2 union lst:
%p A323013     print(lst1):
%p A323013    od:
%Y A323013 Cf. A000045, A002878, A033887, A035312, A036561, A117647.
%K A323013 nonn,tabl
%O A323013 1,2
%A A323013 _Michel Lagneau_, Jan 02 2019