A031878 -id:A031878 - cp's OEIS frontend

A123684 Alternate A016777(n) with A000027(n).

1, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100, 34, 103, 35, 106, 36

Offset: 1

Alford Arnold, Oct 11 2006

a(n) is a diagonal of Table A123685.
The arithmetic average of the first n terms gives the positive integers repeated (A008619). - Philippe Deléham, Nov 20 2013
Images under the modified '3x-1' map: a(n) = n/2 if n is even, (3n-1)/2 if n is odd. (In this sequence, the numbers at even indices n are n/2 [A000027], and the numbers at odd indices n are 3((n-1)/2) + 1 [A016777] = (3n-1)/2.) The latter correspondence interestingly mirrors an insight in David Bařina's 2020 paper (see below), namely that 3(n+1)/2 - 1 = (3n+1)/2. - Kevin Ge, Oct 30 2024

			The natural numbers begin 1, 2, 3, ... (A000027), the sequence 3*n + 1 begins 1, 4, 7, 10, ... (A016777), therefore A123684 begins 1, 1, 4, 2, 7, 3, 10, ...
1/1 = 1, (1+1)/2 = 1, (1+1+4)/3 = 2, (1+1+4+2)/4 = 2, ... - _Philippe Deléham_, Nov 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
David Bařina, Convergence verification of the Collatz problem

Cf. A000027, A008619, A016777, A016825, A031878, A052928, A064455.
Cf. A071045, A092530, A093005, A105638, A123685, A137501, A225126.
Cf. A014682 (modified Collatz map)

import Data.List (transpose)
a123684 n = a123684_list !! (n-1)
a123684_list = concat $ transpose [a016777_list, a000027_list]
-- Reinhard Zumkeller, Apr 29 2013

&cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007

/* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 2013

A123684:=n->n-1/4-(1/2*n-1/4)*(-1)^n: seq(A123684(n), n=1..70); # Wesley Ivan Hurt, Jul 26 2014

CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x,0,70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)
LinearRecurrence[{0,2,0,-1},{1,1,4,2},80] (* Harvey P. Dale, Apr 14 2025 *)

print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))) \\ Klaus Brockhaus, May 12 2007

print(vector(72, n, n-1/4-(1/2*n-1/4)*(-1)^n)); \\ Klaus Brockhaus, May 12 2007

[(n + (2*n-1)*(n%2))//2 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From Klaus Brockhaus, May 12 2007: (Start)
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1+x)^2).
a(n) = (1/4)*(4*n - 1 - (2*n - 1)*(-1)^n).
a(2n-1) = A016777(n-1) = 3(n-1) + 1.
a(2n) = A000027(n) = n.
a(n) = A071045(n-1) + 1.
a(n) = A093005(n) - A093005(n-1) for n > 1.
a(n) = A105638(n+2) - A105638(n+1) for n > 1.
a(n) = A092530(n) - A092530(n-1) - 1.
a(n) = A031878(n+1) - A031878(n) - 1. (End)
a(2*n+1) + a(2*n+2) = A016825(n). - Paul Curtz, Mar 09 2011
a(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011
From Jaroslav Krizek, Mar 22 2011 (Start):
a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.
a(n) = A064455(n) - A137501(n).
Abs(a(n) - A064455(n)) = A052928(n). (End)
a(n) = A225126(n) for n > 1. - Reinhard Zumkeller, Apr 29 2013
a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013
a(n) = n + floor(n/2) for odd n; a(n) = n/2 for even n. - Reinhard Muehlfeld, Jul 25 2014

More terms from Klaus Brockhaus, May 12 2007

A031940 Length of longest legal domino snake using full set of dominoes up to [n:n].

Original entry on oeis.org

1, 3, 6, 9, 15, 19, 28, 33, 45, 51, 66, 73, 91, 99, 120, 129, 153, 163, 190, 201, 231, 243, 276, 289, 325, 339, 378, 393, 435, 451, 496, 513, 561, 579, 630, 649, 703, 723, 780, 801, 861, 883, 946, 969, 1035, 1059, 1128, 1153, 1225, 1251, 1326, 1353, 1431, 1459

Offset: 1

Colin Mallows

nonn

			E.g., for n=4 [ 1:1 ][ 1:2 ][ 2:2 ][ 2:3 ][ 3:3 ][ 3:1 ][ 1:4 ][ 4:4 ][ 4:2 ].

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A031878, A204556.

[((-1)^n*(2 - n) + (2 + n + 2*n^2))/4: n in [1..60]]; // G. C. Greubel, Jun 15 2018

Rest[CoefficientList[Series[x*(1 + 2*x + x^2 - x^3 + x^4)/((1 + x)^2*(1 - x)^3), {x, 0, 50}], x]] (* or *) Table[((-1)^n*(2-n) + (2+n+2*n^2))/4, {n,1, 50}] (* G. C. Greubel, Jun 15 2018 *)

for(n=1, 60, print1(((-1)^n*(2 - n) + (2 + n + 2*n^2))/4, ", ")) \\ G. C. Greubel, Jun 15 2018

Vec(-x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ) + O(x^60)) \\ Felix Fröhlich, Jun 18 2018

C(n, 2) + n if n odd, C(n, 2) + n/2 + 1 if n even. - T. D. Noe, Nov 09 2006
a(n) = A204556(n+1) / (n+1). - Reinhard Zumkeller, Jan 18 2012
G.f.: -x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 13 2012
a(n) = ((-1)^n*(2 - n) + (2 + n + 2*n^2))/4. - G. C. Greubel, Jun 15 2018

Corrected by T. D. Noe, Nov 09 2006

More terms from Felix Fröhlich, Jun 18 2018

A317612 For k >= 1, fill a k X k square with the numbers 1 to k^2 by rows left to right and top to bottom; then read the square by a square clockwise spiral beginning at the top left and spiraling inwards.

Original entry on oeis.org

1, 1, 2, 4, 3, 1, 2, 3, 6, 9, 8, 7, 4, 5, 1, 2, 3, 4, 8, 12, 16, 15, 14, 13, 9, 5, 6, 7, 11, 10, 1, 2, 3, 4, 5, 10, 15, 20, 25, 24, 23, 22, 21, 16, 11, 6, 7, 8, 9, 14, 19, 18, 17, 12, 13, 1, 2, 3, 4, 5, 6, 12, 18, 24, 30, 36, 35, 34, 33, 32, 31, 25, 19, 13, 7, 8, 9, 10, 11, 17, 23, 29, 28, 27, 26, 20, 14, 15, 16, 22, 21, 1, 2, 3, 4, 5, 6, 7, 14, 21, 28, 35, 42, 49, 48, 47, 46, 45, 44, 43

Offset: 1

George E. Laham II and Robert G. Wilson v, Aug 01 2018

Inspired by A317186.

The final term in the k X k spiral is A031878(k+1).

			  1 => 1;
.
  1---2
      |
  3---4  =>  1, 2, 4, 3;
.
  1---2---3
          |
  4---5   6
  |       |
  7---8---9  =>  1, 2, 3, 6, 9, 8, 7, 4, 5;
.
   1---2---3---4
               |
   5---6---7   8
   |       |   |
   9  10--11  12
   |           |
  13--14--15--16
.
  => 1, 2, 3, 4, 8, 12, 16, 15, 14, 13, 9, 5, 6, 7, 11, 10;
.
   1---2---3---4---5
                   |
   6---7---8---9  10
   |           |   |
  11  12--13  14  15
   |   |       |   |
  16  17--18--19  20
   |               |
  21--22--23--24--25
.
  => 1, 2, 3, 4, 5, 10, 15, 20, 25, 24, 23, 22, 21, 16, 11, 6, 7, 8, 9, 14, 19, 18, 17, 12, 13;
.
   1---2---3---4---5---6
                       |
   7---8---9--10--11  12
   |               |   |
  13  14--15--16  17  18
   |   |       |   |   |
  19  20  21--22  23  24
   |   |           |   |
  25  26--27--28--29  30
   |                   |
  31--32--33--34--35--36
.
  => 1, 2, 3, 4, 5, 6, 12, 18, 24, 30, 36, 35, 34, 33, 32, 31, 25, 19, 13, 7, 8, 9, 10, 11, 17, 23, 29, 28, 27, 26, 20, 14, 15, 16, 22, 21;

Cf. A031878, A317186.

(* To form an n X n square table which begins left to right, then top to bottom *) a[i_, j_, n_] := j + n*(i - 1); f[n_] := Table[ a[i, j, n], {i, n}, {j, n}]

cp's OEIS Frontend

A123684 Alternate A016777(n) with A000027(n).

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A031940 Length of longest legal domino snake using full set of dominoes up to [n:n].

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Magma

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A317612 For k >= 1, fill a k X k square with the numbers 1 to k^2 by rows left to right and top to bottom; then read the square by a square clockwise spiral beginning at the top left and spiraling inwards.

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