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

Author

Clark Kimberling, Oct 28 2012

Keywords

• nonn
• tabl

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A000045, A214978, A214984, A214986.

Programs

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}]]