A002517 -id:A002517 - cp's OEIS frontend

A007494 Numbers that are congruent to 0 or 2 mod 3.

0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

The map n -> a(n) (where a(n) = 3n/2 if n even or (3n+1)/2 if n odd) was studied by Mahler, in connection with "Z-numbers" and later by Flatto. One question was whether, iterating from an initial integer, one eventually encountered an iterate = 1 (mod 4). - Jeff Lagarias, Sep 23 2002
Partial sums of 0,2,1,2,1,2,1,2,1,... . - Paul Barry, Aug 18 2007
a(n) = numbers k such that antiharmonic mean of the first k positive integers is not integer. A169609(a(n-1)) = 3. See A146535 and A169609. Complement of A016777. - Jaroslav Krizek, May 28 2010
Range of A173732. - Reinhard Zumkeller, Apr 29 2012
Number of partitions of 6n into two odd parts. - Wesley Ivan Hurt, Nov 15 2014
Numbers m such that 3 divides A000217(m). - Bruno Berselli, Aug 04 2017
Maximal length of a snake like polyomino that fits in a 2 X n rectangle. - Alain Goupil, Feb 12 2020

L. Flatto, Z-numbers and beta-transformations, in Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1002.
Attila Máder, The Use of Experimental Mathematics in the Classroom, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
Kurt Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc., Vol. 8 (1968), pp. 313-321.
P. Sabinin and M. G. Stone, Transforming n-gons by Folding the Plane, Amer. Math. Monthly, Vol. 102, No. 7 (1995), pp. 620-627.
Eric Weisstein's World of Mathematics, Folding.
Robert G. Wilson v, Notes with attachment.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A001651, A032766, A035361, A063574, A132462.
Complement of A016777.
Range of A002517.
Cf. A274406. [Bruno Berselli, Jun 26 2016]

a007494 =  flip div 2 . (+ 1) . (* 3) -- Reinhard Zumkeller, Dec 12 2014

[(6*n+1)/4-(-1)^n/4: n in [0..80]]; // Vincenzo Librandi, Aug 20 2011

a[0]:=0:a[1]:=2:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..71); # Zerinvary Lajos, Mar 16 2008
A007494:=n->floor((3*n+1)/2); seq(A007494(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013

Flatten[{#,#+2}&/@(3Range[0,40])] (* Harvey P. Dale, May 15 2011 *)
Table[2n - Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)

a(n)=n+(n+1)>>1 \\ Charles R Greathouse IV, Jul 25 2011

a(n) = 3*n/2 if n even, otherwise (3*n+1)/2.
If u(1)=0, u(n) = n + floor(u(n-1)/3), then a(n-1) = u(n). - Benoit Cloitre, Nov 26 2002
G.f.: x*(x+2)/((1-x)^2*(1+x)). - Ralf Stephan, Apr 13 2002
a(n) = 3*floor(n/2) + 2*(n mod 2) = A032766(n) + A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = (6*n+1)/4 - (-1)^n/4; a(n) = Sum_{k=0..n-1} (1 + (-1)^(k/2)*cos(k*Pi/2)). - Paul Barry, Aug 18 2007
A145389(a(n)) <> 1. - Reinhard Zumkeller, Oct 10 2008
a(n) = A002943(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = 3*n - a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k>=0} A030308(n,k)*A042950(k). - Philippe Deléham, Oct 17 2011
a(n) = n + ceiling(n/2). - Arkadiusz Wesolowski, Sep 18 2012
a(n) = 2n - floor(n/2) = floor((3n+1)/2) = n + (n + (n mod 2))/2. - Wesley Ivan Hurt, Oct 19 2013
a(n) = A000217(n+1) - A099392(n+1). - Bui Quang Tuan, Mar 27 2015
a(n) = n + floor(n/2) + (n mod 2). - Bruno Berselli, Apr 04 2016
a(n) = Sum_{i=1..n} numerator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k)+(-1)^(k-i). - Wesley Ivan Hurt, Sep 20 2017
E.g.f.: (3*exp(x)*x + sinh(x))/2. - Stefano Spezia, Feb 11 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 04 2021

A002516 Earliest sequence with a(a(n)) = 2n.

Original entry on oeis.org

0, 3, 6, 2, 12, 7, 4, 10, 24, 11, 14, 18, 8, 15, 20, 26, 48, 19, 22, 34, 28, 23, 36, 42, 16, 27, 30, 50, 40, 31, 52, 58, 96, 35, 38, 66, 44, 39, 68, 74, 56, 43, 46, 82, 72, 47, 84, 90, 32, 51, 54, 98, 60, 55, 100, 106, 80, 59, 62, 114, 104, 63, 116, 122, 192, 67, 70, 130

Offset: 0

Colin Mallows

T. D. Noe, Table of n, a(n) for n = 0..1000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences of the a(a(n)) = 2n family

Cf. A002517, A004767, A007379, A017089, A091067.

import Data.List (transpose)
a002516 n = a002516_list !! n
a002516_list = 0 : concat (transpose
[a004767_list, f a002516_list, a017089_list, g $ drop 2 a002516_list])
where f [z] = []; f (_:z:zs) = 2 * z : f zs
g [z] = [z]; g (z:_:zs) = 2 * z : g zs
-- Reinhard Zumkeller, Jun 08 2015

a[0] = 0; a[n_ /; Mod[n, 2] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := n+2; a[n_ /; Mod[n, 4] == 3] := 2(n-2); Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Feb 06 2012, after Henry Bottomley *)

v2(n)=valuation(n,2)
a(n)=2^v2(n)*(-1+3/2*n/2^v2(n)-(-3+1/2*n/2^v2(n))*(-1)^((n/2^v2(n)-1)/2))

a(n)=local(t); if(n<1,0,if(n%2==0,2*a(n/2),t=(n-1)/2; 3*t+1/2-(t-5/2)*(-1)^t)) \\ Ralf Stephan, Feb 22 2004

a(4n) = 2*(a(2n)), a(4n+1) = 4n+3, a(4n+2) = 2*(a(2n+1)), a(4n+3) = 8n+2. - Henry Bottomley, Apr 27 2000
From Ralf Stephan, Feb 22 2004: (Start)
a(n) = n + 2*A006519(n) if odd part of n is of form 4k+1, or 2n - 4*A006519(n) otherwise.
a(2n) = 2*a(n), a(2n+1) = 2n + 3 + (2n - 5)*[n mod 2].
G.f.: Sum_{k>=0} 2^k*t(6t^6 + t^4 + 2t^2 + 3)/(1 - t^4)^2, t = x^2^k. (End)

A007379 Earliest sequence with a(a(n)) = 4n.

Original entry on oeis.org

0, 2, 4, 5, 8, 12, 7, 24, 16, 10, 36, 13, 20, 44, 15, 56, 32, 18, 68, 21, 48, 76, 23, 88, 28, 26, 100, 29, 96, 108, 31, 120, 64, 34, 132, 37, 40, 140, 39, 152, 144, 42, 164, 45, 52, 172, 47, 184, 80, 50, 196, 53, 176, 204, 55, 216, 60, 58, 228, 61, 224, 236, 63, 248

Offset: 0

Colin Mallows, N. J. A. Sloane, Mira Bernstein

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

Cf. A002516, A002517.

a[n_] := a[n] = Which[ Mod[n, 8] == 0, 4*a[n/4], Mod[n, 8] == 1, n+1, Mod[n, 8] == 2, 4*(n-2)+4, Mod[n, 8] == 3, n+2, Mod[n, 8] == 4, 4*a[(n-4)/4+1], Mod[n, 8] == 5, 4*(n-5) + 12, Mod[n, 8] == 6, n+1, True, 4*(n-7)+24]; a[0] = 0; Table[ a[n], {n, 0, 63}] (* Jean-François Alcover, Sep 24 2012 *)

a(8n)=4*a(2n), a(8n+1)=8n+2, a(8n+2)=32n+4, a(8n+3)=8n+5, a(8n+4)=4*a(2n+1), a(8n+5)=32n+12, a(8n+6)=8n+7, a(8n+7)=32n+24

Formula and more terms from Henry Bottomley, Apr 27 2000

A002518 Earliest sequence with a(a(n))=5n.

Original entry on oeis.org

0, 2, 5, 4, 15, 10, 7, 30, 9, 40, 25, 12, 55, 14, 65, 20, 17, 80, 19, 90, 75, 22, 105, 24, 115, 50, 27, 130, 29, 140, 35, 32, 155, 34, 165, 150, 37, 180, 39, 190, 45, 42, 205, 44, 215, 200, 47, 230, 49, 240, 125, 52, 255, 54, 265, 60, 57, 280, 59, 290, 275, 62, 305

Offset: 0

Colin Mallows

Cf. A002516, A002517, A007379.

a[n_] := a[n] = Which[ Mod[n, 5] == 0, 5*a[n/5], Mod[n, 5] == 1, n+1, Mod[n, 5] == 2, 5*(n-2)+5, Mod[n, 5] == 3, n+1, True, 5*(n-4)+15]; a[0] = 0; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

a(5n)=5*a(n), a(5n+1)=5n+2, a(5n+2)=25n+5, a(5n+3)=5n+4, a(5n+4)=25n+15

Corrected description and more terms from Henry Bottomley, Apr 27 2000

A054786 Earliest sequence with a(a(n)) = 6n.

Original entry on oeis.org

0, 2, 6, 4, 18, 7, 12, 30, 9, 48, 11, 60, 36, 14, 78, 16, 90, 19, 24, 102, 21, 120, 23, 132, 108, 26, 150, 28, 162, 31, 42, 174, 33, 192, 35, 204, 72, 38, 222, 40, 234, 43, 180, 246, 45, 264, 47, 276, 54, 50, 294, 52, 306, 55, 288, 318, 57, 336, 59, 348, 66, 62, 366, 64

Offset: 0

Henry Bottomley, Apr 27 2000

Cf. A002516, A002517, A002518, A007379.

a[0] = 0; a[n_] := a[n] = Switch[ Mod[n, 12], 0 | 6, 6*a[n/6], 1 | 3 | 8 | 10, n+1, 2 | 4 | 9 | 11, 6*n-6, 5, n+2, 7, 6*n-12]; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Dec 20 2011, after formula *)

a(12n)=6*a(2n), a(12n+1)=12n+2, a(12n+2)=72n+6, a(12n+3)=12n+4, a(12n+4)=72n+18, a(12n+5)=12n+7, a(12n+6)=6*a(2n+1), a(12n+7)=72n+30, a(12n+8)=12n+9, a(12n+9)=72n+48, a(12n+10)=12n+11, a(12n+11)=72n+60.

Typo in formula corrected by Reinhard Zumkeller, Jul 23 2010

A054787 Earliest sequence with a(a(n))=7n.

Original entry on oeis.org

0, 2, 7, 4, 21, 6, 35, 14, 9, 56, 11, 70, 13, 84, 49, 16, 105, 18, 119, 20, 133, 28, 23, 154, 25, 168, 27, 182, 147, 30, 203, 32, 217, 34, 231, 42, 37, 252, 39, 266, 41, 280, 245, 44, 301, 46, 315, 48, 329, 98, 51, 350, 53, 364, 55, 378, 63, 58, 399, 60, 413, 62, 427, 392

Offset: 0

Henry Bottomley, Apr 27 2000

Cf. A002516, A002517, A002518, A007379.

a[n_] := a[n] = Which[ Mod[n, 7] == 0, 7*a[n/7], Mod[n, 7] == 1, n+1, Mod[n, 7] == 2, 7*(n-2)+7, Mod[n, 7] == 3, n+1, Mod[n, 7] == 4, 7*(n-4)+21, Mod[n, 7] == 5, n+1, Mod[n, 7] == 6, 7*(n-6)+35]; a[0] = 0; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Sep 24 2012 *)

a(7n)=7*a(n), a(7n+1)=7n+2, a(7n+2)=49n+7, a(7n+3)=7n+4, a(7n+4)=49n+21, a(7n+5)=7n+6, a(7n+6)=49n+35

A054790 Earliest sequence with a(a(n))=10n.

Original entry on oeis.org

0, 2, 10, 4, 30, 6, 50, 8, 70, 11, 20, 90, 13, 120, 15, 140, 17, 160, 19, 180, 100, 22, 210, 24, 230, 26, 250, 28, 270, 31, 40, 290, 33, 320, 35, 340, 37, 360, 39, 380, 300, 42, 410, 44, 430, 46, 450, 48, 470, 51, 60, 490, 53, 520, 55, 540, 57, 560, 59, 580, 500, 62, 610

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379.

a[0] = 0; a[n_] := Which[m = Mod[n, 20]; m == 0, 10*n-100, m == 9, n+2, m == 10, n+10, m == 11, 10*n-20, MemberQ[ {2, 4, 6, 8, 13, 15, 17, 19}, m], 10*n-10, True, n+1]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

A054788 Earliest sequence with a(a(n))=8n.

Original entry on oeis.org

0, 2, 8, 4, 24, 6, 40, 9, 16, 56, 11, 80, 13, 96, 15, 112, 64, 18, 136, 20, 152, 22, 168, 25, 32, 184, 27, 208, 29, 224, 31, 240, 192, 34, 264, 36, 280, 38, 296, 41, 48, 312, 43, 336, 45, 352, 47, 368, 320, 50, 392, 52, 408, 54, 424, 57, 72, 440, 59, 464, 61, 480, 63, 496

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379.

nmax = 63; amax = 8*nmax; t = {{0, a[0] = 0}, {1, a[1] = 2}, {2, a[2]}}; While[ !FreeQ[t, a], t = Table[{n, a[n]}, {n, 0, nmax}]; n = Select[t, !IntegerQ[ #[[2]] ] &, 1][[1, 1]]; t2 = Union[ Flatten[ Append[ Select[ t, IntegerQ[ #[[2]] ] &], n]]]; an = If[n == 2, 8, Select[ Complement[ Range[ Max[t2] ], t2], Mod[#, 8] != 0 &, 1][[1]] ]; a[n] = an; While[ an < amax, an = a[n = an] = 8 n]]; Table[ a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 11 2012 *)

A054789 Earliest sequence with a(a(n)) = 9n.

Original entry on oeis.org

0, 2, 9, 4, 27, 6, 45, 8, 63, 18, 11, 90, 13, 108, 15, 126, 17, 144, 81, 20, 171, 22, 189, 24, 207, 26, 225, 36, 29, 252, 31, 270, 33, 288, 35, 306, 243, 38, 333, 40, 351, 42, 369, 44, 387, 54, 47, 414, 49, 432, 51, 450, 53, 468, 405, 56, 495, 58, 513, 60, 531, 62, 549

Offset: 0

Henry Bottomley, Apr 27 2000

Index entries for sequences of the a(a(n)) = 2n family

Cf. A002516, A002517, A002518, A007379, A008585.

a[0] = 0; a[n_] := Which[m = Mod[n, 18]; m == 0, 9*n-81, m == 9, n+9, MemberQ[ {1, 3, 5, 7, 10, 12, 14, 16}, m], n+1, True, 9*n-9]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

cp's OEIS Frontend

A007494 Numbers that are congruent to 0 or 2 mod 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A002516 Earliest sequence with a(a(n)) = 2n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A007379 Earliest sequence with a(a(n)) = 4n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A002518 Earliest sequence with a(a(n))=5n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A054786 Earliest sequence with a(a(n)) = 6n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A054787 Earliest sequence with a(a(n))=7n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A054790 Earliest sequence with a(a(n))=10n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A054788 Earliest sequence with a(a(n))=8n.

Views

Author

Keywords