A316939 - OEIS

%I #27 Sep 13 2021 05:37:32

%S 1,2,2,3,4,3,5,7,7,5,8,12,14,12,8,13,20,26,26,20,13,21,33,46,52,46,33,

%T 21,34,54,79,98,98,79,54,34,55,88,133,177,196,177,133,88,55,89,143,

%U 221,310,373,373,310,221,143,89,144,232,364,531,683,746,683,531,364,232,144,233,376,596,895,1214,1429

%N Triangle read by rows formed using Pascal's rule except that n-th row begins and ends with Fibonacci(n+2).

%e Triangle begins:

%e    1;

%e    2,  2;

%e    3,  4,   3;

%e    5,  7,   7,   5;

%e    8, 12,  14,  12,   8;

%e   13, 20,  26,  26,  20,  13;

%e   21, 33,  46,  52,  46,  33,  21;

%e   34, 54,  79,  98,  98,  79,  54, 34;

%e   55, 88, 133, 177, 196, 177, 133, 88, 55;

%e   ...

%p f:= proc(n,k) option remember;

%p   if k=0 or k=n then combinat:-fibonacci(n+2) else procname(n-1,k)+procname(n-1,k-1) fi

%p end proc:

%p for n from 0 to 10 do

%p   seq(f(n,k),k=0..n)

%p od; # _Robert Israel_, Sep 20 2018

%t t={}; Do[r={}; Do[If[k==0||k==n, m=Fibonacci[n + 2], m=t[[n, k]] + t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t // Flatten

%Y Cf. A316528 (row sums).

%Y Columns k=0..2: A000045, A000071, A001924.

%Y Other Fibonacci borders: A074829, A108617, A316938.

%K nonn,tabl

%O 0,2

%A _Vincenzo Librandi_, Jul 28 2018

%E Incorrect g.f. removed by _Georg Fischer_, Feb 18 2020