cp's OEIS Frontend

A214987 Power round array for the golden ratio, by antidiagonals.

%I A214987 #16 Oct 28 2024 05:12:16
%S A214987 1,1,1,1,2,1,1,3,3,1,1,5,8,4,1,1,8,21,17,7,1,1,13,55,72,48,11,1,1,21,
%T A214987 144,305,329,122,18,1,1,34,377,1292,2255,1353,323,29,1,1,55,987,5473,
%U A214987 15456,15005,5796,842,47,1,1,89,2584,23184,105937,166408,104005
%N A214987 Power round array for the golden ratio, by antidiagonals.
%C A214987 The term "power round sequence" (after "power ceiling sequence" at A214986) extends to sequences generated by recurrences P(n) = round(x*P(n-1)) + g(n), and "power round functions" f(x) to the limit of P(n)/x^n in case x>1 and g(n)/x^n -> 0.  Suppose that h is a nonnegative integer and g(n) is a constant.  If x is a positive integer power of the golden ratio r, then f(x), in many cases, lies in the field Q(sqrt(5)).  Examples matching rows of A214987, using g(n) = 0, follow:
%C A214987 ...
%C A214987 x ... P . .. . . f(x)
%C A214987 r ... A000045 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)
%C A214987 r^2 . A001906 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)
%C A214987 r^3 . A001076 .. 1/2 + sqrt(5)/5 = 0.9472...
%C A214987 r^4 . A004187 .. 1/2 + 7*sqrt(5)/30 = 1.0217...
%C A214987 In general, f(r^k) = 1/2 + sqrt(5)*L(k)/(10*F(k)) for k>1, where L = A000032 (Lucas numbers) and F = A000045 (Fibonacci numbers).
%C A214987 (row 2 of A214987) = (row 1 of A213978 except for its initial 1)
%C A214987 (row n of A214987) = (row n-1 of A213978 for n>2).
%H A214987 Clark Kimberling, <a href="/A214987/b214987.txt">Antidiagonals n = 1..35, flattened</a>
%e A214987 1...1...1....1.....1......1
%e A214987 1...2...3....5.....8......13
%e A214987 1...3...8....21....5......144
%e A214987 1...4...17...72....305....1292
%e A214987 1...7...48...329...2255...15456
%t A214987 r = GoldenRatio;
%t A214987 s[x_, 0] := 1; s[x_, n_] := Round[x*s[x, n - 1]];
%t A214987 t = TableForm[Table[s[r^m, n], {m, 0, 10}, {n, 0, 10}]  ]
%t A214987 u = Flatten[Table[s[r^m, n - m], {n, 0, 10}, {m, 0, n}]]
%Y A214987 Cf. A000045, A214978, A214984, A214986.
%K A214987 nonn,tabl
%O A214987 1,5
%A A214987 _Clark Kimberling_, Oct 28 2012