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A124507 a(n) = floor(exp(n*Pi/2)). 1
1, 4, 23, 111, 535, 2575, 12391, 59609, 286751, 1379410, 6635623, 31920519, 153552935, 738662922, 3553321280, 17093171648, 82226315585, 395547831244, 1902773895292, 9153250784394, 44031505860632, 211812562992413, 1018919543279304, 4901489415968642 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In the complex plane, multiplication by i=sqrt(-1) is equivalent to a rotation by 90 degrees (Pi/2 radians) counterclockwise. If one raises the number e=2.718... to the angle in question (i.e., Pi/2, which corresponds to the complex number i), the result is a number 4.81047738... which can be thought of as corresponding (but not being equivalent to) the imaginary unit, i. The next 90-degree sweep, which corresponds to i^2, gives e^Pi = 23.1406... . Continuing around the circle, we find the numbers e^(3*Pi/2) = 111.31..., e^(2*Pi) = 535.4..., etc. Truncating these numbers to integers yields the sequence 1, 4, 23, 111, etc. I came across this sequence as I tried to find a real number that corresponded to the complex number 'i' in the complex plane.

REFERENCES

Frank Ayres, Jr., Robert E. Moyer and Murray R. Spiegel, "Essential Trigonometry and Algebra for University Students" (?) (1999), p. 3.

Roger Penrose, The Road to Reality (?) (2005), p. 88 (figure 5.3)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Euler's Famous Identity.

FORMULA

a(n) = floor(exp(n*Pi/2)).

MAPLE

Digits:= 2000:

a:= n-> floor(exp(n*Pi/2)):

seq(a(n), n=0..30);  # Alois P. Heinz, Nov 25 2018

MATHEMATICA

Table[ Floor@ Exp[n*Pi/2], {n, 0, 21}] (* Robert G. Wilson v *)

PROG

(PARI) vector(30, n, n--; floor(exp(n*Pi/2))) \\ G. C. Greubel, Nov 25 2018

(MAGMA) R:= RealField(10); [Floor(Exp(n*Pi(R)/2)): n in [0..30]]; // G. C. Greubel, Nov 25 2018

(Sage) [floor(exp(n*pi/2)) for n in range(30)] # G. C. Greubel, Nov 25 2018

CROSSREFS

Cf. A042972.

Sequence in context: A197868 A017973 A306669 * A239813 A174248 A297309

Adjacent sequences:  A124504 A124505 A124506 * A124508 A124509 A124510

KEYWORD

nonn

AUTHOR

Zacariaz Martinez, Dec 27 2006

EXTENSIONS

Edited and extended by Robert G. Wilson v, Dec 31 2006

Comments edited by Jon E. Schoenfield, Nov 25 2018

STATUS

approved

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Last modified August 26 05:23 EDT 2019. Contains 326328 sequences. (Running on oeis4.)