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A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).
(Formerly M2308)
8
3, 4, 5, 7, 10, 14, 20, 29, 43, 64, 95, 142, 212, 317, 475, 712, 1067, 1600, 2399, 3598, 5396, 8093, 12139, 18208, 27311, 40966, 61448, 92171, 138256, 207383, 311074, 466610, 699914, 1049870, 1574804, 2362205, 3543307, 5314960, 7972439, 11958658, 17937986, 26906978 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This sequence was originally defined in Popular Computing in 1974 by a sieve, as follows. Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every third term. Repeat, always crossing off every third term of those that remain. The numbers that are left form the sequence. The recurrence given here for the sequence was found by Colin Mallows. The problem asked for the 1000-th term, and was unsolved for several years.

REFERENCES

Popular Computing (Calabasas, CA), Problem 43, Sieves, sieve #5, Vol. 2 (No. 13, Apr 1974), pp. 6-7; Vol. 2 (No. 17, Aug 1974), page 16; Vol. 5 (No. 51, Jun 1977), p. 17.

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

LINKS

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..1000  [First 500 terms from T. D. Noe]

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #5. [Annotated and scanned copy]

H. P. Robinson, C. L. Mallows, & N. J. A. SloaneCorrespondence, 1975

Index entries for sequences generated by sieves

EXAMPLE

The first few sieving stages are as follows:

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...

3 4 5 X 7 8 X 10 11 XX 13 14 XX 16 17 XX 19 20 ...

3 4 5 X 7 X X 10 11 XX XX 14 XX 16 XX XX 19 20 ...

3 4 5 X 7 X X 10 XX XX XX 14 XX 16 XX XX XX 20 ...

3 4 5 X 7 X X 10 XX XX XX 14 XX XX XX XX XX 20 ...

MAPLE

f:=proc(n) option remember; if n=1 then RETURN(3) fi; f(n-1)+floor( (f(n-1)-1)/2 ); end;

MATHEMATICA

NestList[#+Floor[(#-1)/2]&, 3, 50]  (* Harvey P. Dale, Mar 18 2011 *)

PROG

(PARI) v=vector(100); v[1]=3; for(n=2, #v, v[n]=floor((3*v[n-1]-1)/2)); v \\ Clark Kimberling, Dec 30 2010

(Haskell)

a003312 n = a003312_list !! (n-1)

a003312_list = sieve [3..] where

   sieve :: [Integer] -> [Integer]

   sieve (x:xs) = x : (sieve $ xOff xs)

   xOff :: [Integer] -> [Integer]

   xOff (x:x':_:xs) = x : x': (xOff xs)

-- Reinhard Zumkeller, Feb 21 2011

(Python)

from sympy import floor

l=[0, 3]

for n in xrange(2, 101): l+=[l[n - 1] + floor((l[n - 1] - 1)/2), ]

print l[1:] # Indranil Ghosh, Jun 09 2017

CROSSREFS

Cf. A003309, A003310, A100464, A100562, A006999, A061418, A070885, A003311.

Sequence in context: A236384 A073957 A162311 * A022440 A088130 A046840

Adjacent sequences:  A003309 A003310 A003311 * A003313 A003314 A003315

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised by N. J. A. Sloane, Dec 01 2004 and May 10 2015

STATUS

approved

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.