This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342689 #11 Dec 23 2024 03:14:53
%S A342689 0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,3,3,1,1,0,1,5,7,4,1,1,0,1,8,31,13,5,
%T A342689 1,1,0,1,13,255,121,21,6,1,1,0,1,21,8191,3280,341,31,7,1,1,0,1,34,
%U A342689 2097151,797161,21845,781,43,8,1,1,0,1,55,17179869184,5230176601,22369621,97656,1555,57,9,1,1,0
%N A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).
%C A342689 Replacing Fibonacci(n), A000045, with Lucas(n), A000032, you get another square array B(n,k). The terms satisfy the same recurrence equation B(n,k) = (k - 1) * B(n-1,k) * B(n-2,k) + B(n-1,k) + B(n-2,k) for k >= 0 and n > 1 with initial values B(0,k) = k+1 and B(1,k) = 1. Please take account of lim_{k -> 1} (k^Lucas(n) - 1) / (k - 1) = Lucas(n).
%F A342689 A(n,k) = (k - 1) * A(n-1,k) * A(n-2,k) + A(n-1,k) + A(n-2,k) for k >= 0 and n > 1 with initial values A(0,k) = 0 and A(1,k) = 1.
%e A342689 The array A(n,k) for k >= 0 and n >= 0 begins:
%e A342689 n \ k: 0 1 2 3 4 5 6 7 8 9 10 11
%e A342689 =========================================================================
%e A342689 0 : 0 0 0 0 0 0 0 0 0 0 0 0
%e A342689 1 : 1 1 1 1 1 1 1 1 1 1 1 1
%e A342689 2 : 1 1 1 1 1 1 1 1 1 1 1 1
%e A342689 3 : 1 2 3 4 5 6 7 8 9 10 11 12
%e A342689 4 : 1 3 7 13 21 31 43 57 73 91 111 133
%e A342689 5 : 1 5 31 121 341 781 1555 2801
%e A342689 6 : 1 8 255 3280 21845 97656
%e A342689 7 : 1 13 8191 797161 22369621
%e A342689 8 : 1 21 2097151 5230176601
%e A342689 9 : 1 34 17179869184
%e A342689 10 : 1 55
%e A342689 11 : 1 89
%e A342689 etc.
%Y A342689 Cf. A011655 (column k = -1), A057427 (column 0), A000045 (column 1), A063896 (column 2), A000004 (row 0), A000012 (rows 1, 2), A000027 (row 3), A002061 (row 4), A053699 (row 5), A053717 (row 6), A060887 (row 7).
%K A342689 nonn,easy,tabl
%O A342689 0,12
%A A342689 _Werner Schulte_, May 18 2021