A214987 Power round array for the golden ratio, by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 8, 4, 1, 1, 8, 21, 17, 7, 1, 1, 13, 55, 72, 48, 11, 1, 1, 21, 144, 305, 329, 122, 18, 1, 1, 34, 377, 1292, 2255, 1353, 323, 29, 1, 1, 55, 987, 5473, 15456, 15005, 5796, 842, 47, 1, 1, 89, 2584, 23184, 105937, 166408, 104005

Offset

1,5

Comments

The term "power round sequence" (after "power ceiling sequence" at A214986) extends to sequences generated by recurrences P(n) = round(x*P(n-1)) + g(n), and "power round 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 A214987, using g(n) = 0, follow:

...

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

r ... A000045 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)

r^2 . A001906 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)

r^3 . A001076 .. 1/2 + sqrt(5)/5 = 0.9472...

r^4 . A004187 .. 1/2 + 7*sqrt(5)/30 = 1.0217...

In general, f(r^k) = 1/2 + sqrt(5)*L(k)/(10*F(k)) for k>1, where L = A000032 (Lucas numbers) and F = A000045 (Fibonacci numbers).

(row 2 of A214987) = (row 1 of A213978 except for its initial 1)

(row n of A214987) = (row n-1 of A213978 for n>2).

Links

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

Example

1...1...1....1.....1......1
1...2...3....5.....8......13
1...3...8....21....5......144
1...4...17...72....305....1292
1...7...48...329...2255...15456

Mathematica

r = GoldenRatio;
s[x_, 0] := 1; s[x_, n_] := Round[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, A214986.

Sequence in context: A199333 A089980 A181031 * A203946 A128545 A194672

Adjacent sequences: A214984 A214985 A214986 * A214988 A214989 A214990

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 28 2012

Status

approved