%I #8 Feb 10 2019 01:56:58
%S 1,0,1,0,0,1,1,0,0,1,0,1,0,0,2,1,0,1,0,0,3,1,1,0,1,0,0,5,1,1,1,0,2,0,
%T 0,8,2,1,1,1,0,3,0,0,13,2,2,1,1,2,0,5,0,0,21,3,2,2,1,2,3,0,8,0,0,34,4,
%U 3,2,2,2,3,5,0,13,0,0,55,5,4,3,2,4,3,5
%N Eigentriangle whose left border is the Padovan sequence, and whose right border and row sums are a modified Fibonacci sequence.
%C Left border = A000931, the Padovan sequence: (1, 0, 0, 1, 0, 1, 1, 1, 2, 2, ...).
%C Right border = (1, 1, 1, 1, 2, 3, 5, 8, 13, 21, ...)
%C Row sums = (1, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).
%C Sum of n-th row terms = rightmost term in next row.
%H Nathaniel Johnston, <a href="/A144961/b144961.txt">Rows n = 0..150, flattened</a>
%F Triangle read by rows, termwise products of a Padovan "decrescendo" triangle: (1; 0,1; 0,0,1; 1,0,0,1,...) and the Fibonacci series preceded by two 1's: (1, 1, 1, 1, 2, 3, 5, 8, ...); (i.e., the INVERT transform of the Padovan sequence).
%F a(n,k) = Fibonacci(max(k-1,1))*A000931(n-k). - _Nathaniel Johnston_, May 05 2011
%e First few rows of the triangle:
%e 1;
%e 0, 1;
%e 0, 0, 1;
%e 1, 0, 0, 1;
%e 0, 1, 0, 0, 2;
%e 1, 0, 1, 0, 0, 3;
%e 1, 1, 0, 1, 0, 0, 5;
%e 1, 1, 1, 0, 2, 0, 0, 8;
%e 2, 1, 1, 1, 0, 3, 0, 0, 13;
%e 2, 2, 1, 1, 2, 0, 5, 0, 0, 21;
%e 3, 2, 2, 1, 2, 3, 0, 8, 0, 0, 34;
%e 4, 3, 2, 2, 2, 3, 5, 0, 13, 0, 0, 55;
%e ...
%p with(combinat): A000931 := proc(n) option remember: if(n=0)then return 1: elif(n<=2)then return 0: else return procname(n-2)+procname(n-3): fi: end: A144961 := proc(n,k) return fibonacci(max(k-1,1))*A000931(n-k) end:for n from 0 to 12 do for k from 0 to n do printf("%d, ",A144961(n,k)); od: printf("\n"); od: # _Nathaniel Johnston_, May 05 2011
%Y Cf. A000045, A000931.
%K nonn,easy,tabl
%O 0,15
%A _Gary W. Adamson_, Sep 27 2008
%E Example triangle corrected by _Nathaniel Johnston_, May 05 2011