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A219085 Floor((n + 1/2)^3). 9
0, 3, 15, 42, 91, 166, 274, 421, 614, 857, 1157, 1520, 1953, 2460, 3048, 3723, 4492, 5359, 6331, 7414, 8615, 9938, 11390, 12977, 14706, 16581, 18609, 20796, 23149, 25672, 28372, 31255, 34328, 37595, 41063, 44738, 48627, 52734, 57066 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/3 and { } = fractional part.  In general, suppose that f is a continuous strictly increasing downward concave function, with f(1)>=0 and f(k)+1/2 not an integer.  Let J(k) denote the inequality {f(k)} < 1/2 < {f(k+1)}, where {}= fractional part; equivalently, [{f(k)} + 1/2] = 0 and [{f(k+1)} + 1/2] = 1, where [ ] = floor.  Thus J(k) holds if the integer nearest f(k+1) exceeds the integer nearest f(k), so that k can be regarded as a "jump point for f".  The solutions of J(k) are the numbers [g(n)+1/2)] for n >= 0, where g = (inverse of f).

Conjecture:  if d is a positive integer and f(x) = x^(1/d), then the solutions of J(k) form a linearly recurrent sequence.

This conjecture was proved by David Moews; see Problem 21 in "Unsolved Problems and Rewards".  - Clark Kimberling, Feb 06 2013

Guide to related sequences:

f(x) ....... jump sequence ... linear recurrence order

x^(1/2) .... A002378 ......... 3

x^(1/3) .... A219085 ......... 7

x^(2/3) .... A203302 ......... (not linearly recurrent)

x^(1/4) .... A219086 ......... 5

x^(3/4) .... A219087 ......... (not linearly recurrent)

x^(1/5) .... A219088 ......... 21

x^(1/6) .... A219089 ......... 21

x^(1/7) .... A219090 ..........71

x^(1/8) .... A219091 ......... 23

log(x) ..... A219092 ......... (not linearly recurrent)

log_2(x) ... A084188 ......... (not linearly recurrent)

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Unsolved Problems and Rewards, Problem 21.

FORMULA

a(n) = floor((n + 1/2)^3).

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).

G.f.: (3*x +6*x^2 +6*x^3 +7*x^4 +x^5 +x^6)/(u*v), where u = (1 - x)^4, v = 1 + x + x^2 + x^3.

a(n) = (n+1/2)^3 +(2*i^(n*(n-1))+(-1)^n-4)/8, where i=sqrt(-1). - Bruno Berselli, Dec 21 2012

EXAMPLE

Let p=1/3.  Then

3^p=1.44... and 4^p=1.58..., so 3 is a jump point.

15^p=2.46... and 16^p=2.51..., so 15 is a jump point.

MATHEMATICA

Table[Floor[(n + 1/2)^3], {n, 0, 100}]

PROG

(PARI) a(n)=n^3 + (6*n^2 + 3*n)\4 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A002378, A203302.

Sequence in context: A012222 A069267 A059270 * A093627 A192060 A146805

Adjacent sequences:  A219082 A219083 A219084 * A219086 A219087 A219088

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 20 2012

STATUS

approved

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Last modified August 17 06:13 EDT 2017. Contains 290635 sequences.