A214986 Power ceiling array for the golden ratio, by antidiagonals.

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, 144, 399, 329, 134, 18, 1, 1, 54, 377, 1691, 2255, 1487, 323, 30, 1, 1, 88, 987, 7164, 15456, 16492, 5796, 872, 47, 1, 1, 143, 2584, 30348, 105937, 182900

Offset

1,5

Comments

row 0: A000012 ... row 6: A049660

row 1: A000071 ... row 8: A049668

row 2: A001906 ... col 0: A000012

row 3: A049652 ... col 1: A169986

row 4: A004187

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.

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.

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:

...

x ... P ........ f(x)

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

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

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

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

...

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

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)).

Links

Clark Kimberling, Antidiagonals n = 1..35, flattened

Eric Weisstein's World of Mathematics, Power Ceilings

Formula

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

Example

Northwest corner:
1...1....1.....1......1.......1
1...2....4.....7......12......20
1...3....8.....21.....55......144
1...5....22....94.....399.....1691
1...7....48....329....2255....15456
1...19...134...1487...16492...182900

Mathematica

r = GoldenRatio;
s[x_, 0] := 1; s[x_, n_] := Ceiling[x*s[x, n - 1]];
t = TableForm[Table[s[r^m, n], {m, 0, 10}, {n, 0, 10}]  ]
u = Flatten[Table[s[r^m, n - m], {n, 0, 10}, {m, 0, n}]]

Crossrefs

Cf. A000045, A214978, A214984, A214987.

Sequence in context: A210764 A091186 A138155 * A307133 A218664 A247286

Adjacent sequences: A214983 A214984 A214985 * A214987 A214988 A214989

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 28 2012

Status

approved