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Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) = Fibonacci rabbit sequence number n coded in base k.
12

%I #32 Oct 25 2018 20:36:26

%S 1,1,1,1,2,2,1,3,5,3,1,4,10,22,5,1,5,17,93,181,8,1,6,26,276,2521,5814,

%T 13,1,7,37,655,17681,612696,1488565,21,1,8,50,1338,81901,18105620,

%U 4019900977,12194330294,34,1,9,65,2457,289045,255941280,1186569930001,6409020585966267,25573364166211253,55

%N Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) = Fibonacci rabbit sequence number n coded in base k.

%H Alois P. Heinz, <a href="/A144287/b144287.txt">Antidiagonals n = 1..16, flattened</a>

%H H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/15-4/gould.pdf">Sequences associated with t-ary coding of Fibonacci's rabbits</a>, Fib. Quart., 15 (1977), 311-318.

%F See program.

%e Square array begins:

%e 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, ...

%e 2, 5, 10, 17, 26, ...

%e 3, 22, 93, 276, 655, ...

%e 5, 181, 2521, 17681, 81901, ...

%p f:= proc(n,b) option remember; `if`(n<2, [n,n], [f(n-1, b)[1]*

%p b^f(n-1, b)[2] +f(n-2, b)[1], f(n-1, b)[2] +f(n-2, b)[2]])

%p end:

%p A:= (n,k)-> f(n,k)[1]:

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..11);

%t f[n_, b_] := f[n, b] = If[n < 2, {n, n}, {f[n-1, b][[1]]*b^f[n-1, b][[2]] + f[n-2, b][[1]], f[n-1, b][[2]] + f[n-2, b][[2]]}]; t[n_, k_] := f[n, k][[1]]; Flatten[ Table[t[n, 1+d-n], {d, 1, 11}, {n, 1, d}]] (* _Jean-François Alcover_, translated from Maple, Dec 09 2011 *)

%Y Columns k=1-10 give: A000045, A005203, A005205, A320986, A320987, A320988, A320989, A320990, A320991, A061107 and A036299.

%Y Rows n=1-3 give: A000012, A001477, A002522.

%Y Main diagonal gives A144288.

%K base,nice,nonn,tabl

%O 1,5

%A _Alois P. Heinz_, Sep 17 2008