A214986 - OEIS

%I #10 Feb 16 2025 08:33:18

%S 1,1,1,1,2,1,1,4,3,1,1,7,8,5,1,1,12,21,22,7,1,1,20,55,94,48,12,1,1,33,

%T 144,399,329,134,18,1,1,54,377,1691,2255,1487,323,30,1,1,88,987,7164,

%U 15456,16492,5796,872,47,1,1,143,2584,30348,105937,182900

%N Power ceiling array for the golden ratio, by antidiagonals.

%C row 0: A000012 ... row 6: A049660

%C row 1: A000071 ... row 8: A049668

%C row 2: A001906 ... col 0: A000012

%C row 3: A049652 ... col 1: A169986

%C row 4: A004187

%C For x>1, define c(x,0) = 1 and c(x,n) = ceiling(x*c(x,n-1)) for n>0.  Row m of A214986 is the sequence c(r^m,n), where r = golden ratio = (1 + sqrt(5))/2.  The name of the array corresponds to the power ceiling function f(x) = limit of c(x,n)/x^n as n increases without bound; f(x) generalizes the case for x = 3/2 as described under "Power Ceilings" at MathWorld.  For a graph of f(x), see the Mathematica program at A083286.

%C The term "power ceiling sequence" extends to sequences generated by recurrences P(n) = ceiling(x*P(n-1)) + g(n), and "power ceiling functions" f(x) to the limit of P(n)/x^n in case x>1 and g(n)/x^n -> 0.

%C 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 A214986, using g(n) = 0, follow:

%C ...

%C x ... P ........ f(x)

%C r ... A000071 .. (5 + 2*sqrt(5))/2 = 1.8944... (A010532)

%C r^2 . A001906 .. (5 + 3*sqrt(5))/10 = 1.7082...(A176015)

%C r^3 . A049652 .. (25 + 11*sqrt(5))/40 = 1.2399...

%C r^4 . A004187 .. (15 + 7*sqrt(5))/10 = 1.0219...

%C ...

%C If k is odd, then f(r^k) = r^k((b(k) + c(k))/d(k)), where

%C b(k) = L(j)^2 + L(j-1)^2, where j=[(k+1)/2], L=A000032 (Lucas numbers); c(k) = (L(k)+2)*sqrt(5); d(k) = 10*F(k)*L(k), where F=A000045 (Fibonacci numbers).  If k is even, then f(r^k) = r^k/(F(k)*sqrt(5)).

%H Clark Kimberling, <a href="/A214986/b214986.txt">Antidiagonals n = 1..35, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerCeilings.html">Power Ceilings</a>

%F The odd-numbered rows of A214986 are even-numbered rows of A213978; the even-numbered rows of A214986 are odd-numbered rows of A214984.

%e Northwest corner:

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

%e 1...2....4.....7......12......20

%e 1...3....8.....21.....55......144

%e 1...5....22....94.....399.....1691

%e 1...7....48....329....2255....15456

%e 1...19...134...1487...16492...182900

%t r = GoldenRatio;

%t s[x_, 0] := 1; s[x_, n_] := Ceiling[x*s[x, n - 1]];

%t t = TableForm[Table[s[r^m, n], {m, 0, 10}, {n, 0, 10}]  ]

%t u = Flatten[Table[s[r^m, n - m], {n, 0, 10}, {m, 0, n}]]

%Y Cf. A000045, A214978, A214984, A214987.

%K nonn,tabl,changed

%O 1,5

%A _Clark Kimberling_, Oct 28 2012