login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A214986 Power ceiling array for the golden ratio, by antidiagonals. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 20 14:44 EDT 2021. Contains 347586 sequences. (Running on oeis4.)