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

Author

Clark Kimberling, Oct 28 2012

Keywords

• nonn
• tabl

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)).

Examples

			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
		

Links

• Clark Kimberling, Antidiagonals n = 1..35, flattened
• Eric Weisstein's World of Mathematics, Power Ceilings

Crossrefs

Cf. A000045, A214978, A214984, A214987.

Programs

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}]]

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.