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A003231 a(n) = floor(n*(sqrt(5)+5)/2).
(Formerly M2618)
16
3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 47, 50, 54, 57, 61, 65, 68, 72, 75, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 206, 209 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A003231(n) is the position of n*tau in the ordered union of R and S. The position of n*(tau - 1) is A249115(n). - Clark Kimberling, Oct 21 2014

This is the function named c in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 06 2015

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..58.

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.

Index entries for sequences related to Beatty sequences

FORMULA

a(n) = 2*n + A000201(n). - R. J. Mathar, Aug 22 2014

MAPLE

A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2015

MATHEMATICA

With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)

PROG

(PARI) a(n)=floor(n*(sqrt(5)+5)/2)

(Haskell)

a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral

-- Reinhard Zumkeller, Oct 03 2014

(MAGMA) [Floor(n*(Sqrt(5)+5)/2): n  in [1..100]]; // Vincenzo Librandi, Oct 23 2014

CROSSREFS

Cf. A000201, A003231, A003233, A003234, A249115.

Sequence in context: A189999 A310188 A171983 * A189460 A172323 A190080

Adjacent sequences:  A003228 A003229 A003230 * A003232 A003233 A003234

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description and more terms from Michael Somos, Jun 07 2000

STATUS

approved

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Last modified August 17 17:08 EDT 2019. Contains 326059 sequences. (Running on oeis4.)