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A214986
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Power ceiling array for the golden ratio, by antidiagonals.
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6
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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
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OFFSET
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1,5
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COMMENTS
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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^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)).
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LINKS
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FORMULA
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EXAMPLE
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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
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MATHEMATICA
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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}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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