Hans G. Oberlack - cp's OEIS frontend

A367914 Movement sequence in the counter-clockwise undulating spiral, whereby 1, 2, 3, 4 represent moves to the right, down, left and up.

1, 4, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 1

Hans G. Oberlack, Dec 04 2023

  y ^
    |
  4 |              2---3
    |              |   |
  3 |          2---3   4---3
    |          |           |
  2 |      2---3   2---3   4---3
    |      |       |   |       |
  1 |  2---3   2---3   4---3   4---3
    |  |       |           |       |
  0 |  1---2   1---2   1---4   1---4
    |      |       |           |       |
 -1 |      1---2   1-- 2   1---4   1---4
    |          |       |   |       |
 -2 |          1---2   1---4   1---4
    |              |           |
 -3 |              1---2   1---4
    |                  |   |
 -4 |                  1---4
    +------------------------------------>
      -4  -3  -2  -1   0   1   2   3   4 x

Cf. A359058, A359216, A359217, A063826.

a(k1)=1 with k1=i^2*8+i*0+2*j+1  with i,j >= 0 and j<=4i.
a(k2)=2 with k2=i^2*8+i*12+2*j+6 with i,j >= 0 and j<=4*i+3.
a(k3)=3 with k3=i^2*8+i*8+2*j+3 with  i,j >= 0 and j<=4*i+2.
a(k4)=4 with k4=i^2*8+i*4+2*j+2 with  i,j >= 0 and j<=4*i+1.

A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1

Offset: 0

Hans G. Oberlack, Dec 21 2022

X-coordinates are given in A359216.

			   y ^
     |
   4 |             25--24
     |              |   |
   3 |         27--26  23--22
     |          |           |
   2 |     29--28   5---4  21--20
     |      |       |   |       |
   1 | 31--30   7---6   3---2  19--18
     |  |       |           |       |
   0 | 32--33   8---9   0---1  16--17
     |      |       |           |       |
  -1 |     34--35  10--11  14--15  46--47
     |          |       |   |       |
  -2 |         36--37  12--13  44--45
     |              |           |
  -3 |             38--39  42--43
     |                  |   |
  -4 |                 40--41
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Counterclockwise undulating spiral in a square grid
Rémy Sigrist, PARI program
Index entries for sequences related to coordinates of 2D curves

Cf. A180714, A359058, A359216.

PARI
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See Links section.
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Conjecture: a(n) = T10 + T15 + T20 + T21 where
  T1  = floor(n/16);
  T2  = sqrt(2*T1 + 1/4);
  T3  = floor(T2 - 1/2);
  T4  = n - T3*(T3+1)*16/2;
  T5  = (T3+1)*16;
  T6  = T4 + (3/4)*T5 - 1;
  T7  = T6/T5;
  T8  = floor(T7);
  T9  = 1 - T8;
  T10 = T9 - floor(T4/2);
  T11 = T4 + (2/4)*T5 - 1;
  T12 = T11/T5;
  T13 = floor(T12);
  T14 = T8 - T13;
  T15 = T14*floor((T5 - T11)/2);
  T16 = T4 + (1/4)*T5 - 1;
  T17 = T16/T5;
  T18 = floor(T17);
  T19 = T13 - T18;
  T20 = -T19*floor((T4 - T5/2)/2);
  T21 = -T18*floor((T5 - T4 + 1)/2).
a(2*n) = A180714(n). - Rémy Sigrist, Apr 01 2023

A359216 X-coordinates of a point moving in a counterclockwise undulating spiral in a square grid.

Original entry on oeis.org

0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, -5, -5, -4

Offset: 0

Hans G. Oberlack, Dec 21 2022

Y-coordinates are given in A359217.
The undulating spiral is
   y ^
     |
   4 |             25--24
     |              |   |
   3 |         27--26  23--22
     |          |           |
   2 |     29--28   5---4  21--20
     |      |       |   |       |
   1 | 31--30   7---6   3---2  19--18
     |  |       |           |       |
   0 | 32--33   8---9   0---1  16--17
     |      |       |           |       |
  -1 |     34--35  10--11  14--15  46--47
     |          |       |   |       |
  -2 |         36--37  12--13  44--45
     |              |           |
  -3 |             38--39  42--43
     |                  |   |
  -4 |                 40--41
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Counterclockwise undulating spiral in a square grid
Rémy Sigrist, PARI program
Index entries for sequences related to coordinates of 2D curves

Cf. A329116, A359058, A359217.

PARI
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See Links section.
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a(2*n) = A329116(n). - Rémy Sigrist, Apr 01 2023

A359058 a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 1, 4, 5, 2, 5, 4, 9, 10, 5, 8, 5, 10, 9, 16, 17, 10, 13, 8, 13, 10, 17, 16, 9, 10, 5, 8, 5, 10, 9, 16, 17, 10, 13, 8, 13, 10, 17, 16, 25, 26, 17, 20, 13, 18, 13, 20, 17, 26, 25, 36, 37, 26, 29, 20, 25, 18, 25, 20, 29, 26, 37, 36, 25, 26, 17, 20, 13, 18, 13, 20

Offset: 0

Hans G. Oberlack, Dec 14 2022

The spiral coordinates are A359216 and A359217.

			The spiral begins as follows and for instance point n=7 is at x=-2,y=1 so that a(7) = (-2)^2 + 1^2 = 5.
   y ^
     |
   4 |             17--16
     |              |   |
   3 |         13--10   9--10
     |          |           |
   2 |     13---8   5---4   5---8
     |      |       |   |       |
   1 | 17--10   5---2   1---2   5--10
     |  |       |           |       |
   0 | 16---9   4---1   0---1   4---9
     |      |       |           |       |
  -1 |     10---5   2---1   2---5  10--17
     |          |       |   |       |
  -2 |          8---5   4---5   8--13
     |              |           |
  -3 |             10---9  10--13
     |                  |   |
  -4 |                 16--17
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Undulating counterclockwise spiral in a square grid
Rémy Sigrist, PARI program

Cf. A001481, A336336, A359216, A359217.

PARI
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See Links section.
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a(n) = A359216(n)^2 + A359217(n)^2.

A343203 Sum of x- and y-coordinate of a point moving in a triangular spiral.

Original entry on oeis.org

0, 1, 1, -1, -3, -2, -1, 0, 1, 2, 2, 2, 2, 0, -2, -4, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 1, -1, -3, -5, -7, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 2, 0, -2, -4, -6, -8, -10, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 1, -1, -3, -5, -7, -9, -11, -13, -15

Offset: 0

Hans G. Oberlack, Apr 07 2021

sign

			   y
     |
   4 |                         56
     |                           \
     |                            \
     |                             \
   3 |                         30  55
     |                         / \   \
     |                        /   \   \
     |                       /     \   \
   2 |                     31  12  29  54
     |                     /   / \   \   \
     |                    /   /   \   \   \
     |                   /   /     \   \   \
   1 |                 32  13   2  11  28  53
     |                 /   /   / \   \   \   \
     |                /   /   /   \   \   \   \
     |               /   /   /     \   \   \   \
   0 |             33  14   3   0---1  10  27  52
     |             /   /   /             \   \   \
     |            /   /   /               \   \   \
     |           /   /   /                 \   \   \
  -1 |         34  15   4---5---6---7---8---9  26  51
     |         /   /                             \   \
     |        /   /                               \   \
     |       /   /                                 \   \
  -2 |     35  16--17--18--19--20--21--22--23--24--25  50
     |     /                                             \
     |    /                                               \
     |   /                                                 \
  -3 | 36--37--38--39--40--41--42--43--44--45--46--47--48--49
     |
     +--------------------------------------------------------
   x:  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
Count as follows. Start at n=0 with coordinates (0,0). So first term is 0.
Next step is to move to n=1 with coordinates (1,0) resulting in sum=1. So the sequence begins 0,1,...
Next step is to move to n=2 with coordinates (0,1) resulting in sum=1. So the sequence begins 0,1,1,...
Next step is to move to n=3 with coordinates (-1,0) resulting in sum=-1. So the sequence begins 0,1,1,1,...
Next step is to move to n=4 with coordinates (-2,-1) resulting in sum=-3. So the sequence begins 0,1,1,-1,-3,...
and so on.

Cf. A329972, A329116.

A322465 Numbers on the 0-9-10-line in a spiral on an equilateral triangular lattice.

Original entry on oeis.org

0, 9, 10, 31, 32, 65, 66, 111, 112, 169, 170, 239, 240, 321, 322, 415, 416, 521, 522, 639, 640, 769, 770, 911, 912, 1065, 1066, 1231, 1232, 1409, 1410, 1599, 1600, 1801, 1802, 2015, 2016, 2241, 2242, 2479, 2480, 2729, 2730, 2991, 2992, 3265, 3266, 3551, 3552

Offset: 0

Hans G. Oberlack, Dec 09 2018

Sequence found by reading the line from 0, in the direction 0, 9, 10, ... in the triangle spiral.

Colin Barker, Table of n, a(n) for n = 0..1000
Hans G. Oberlack, Triangle spiral line 0-9-10
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Bisection (even part) gives A202804.

a:= n-> `if`(n::even, n*((3/2)*n+2), (n+1)*((3/2)*(n+1)+2)-1): seq(a(n), n=0..50); # Muniru A Asiru, Dec 20 2018

concat(0, Vec(x*(9 + x + 3*x^2 - x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 18 2018

For even n: a(n) = n*((3/2)*n+2).
For odd n: a(n) = a(n+1)-1 = (n+1)*((3/2)*(n+1)+2)-1.
From Colin Barker, Dec 18 2018: (Start)
G.f.: x*(9 + x + 3*x^2 - x^3) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)

A322462 Numbers on the 0-1-12 line in a spiral on a grid of equilateral triangles.

Original entry on oeis.org

0, 1, 12, 13, 36, 37, 72, 73, 120, 121, 180, 181, 252, 253, 336, 337, 432, 433, 540, 541, 660, 661, 792, 793, 936, 937, 1092, 1093, 1260, 1261, 1440, 1441, 1632, 1633, 1836, 1837, 2052, 2053, 2280, 2281, 2520, 2521, 2772, 2773, 3036, 3037, 3312, 3313, 3600

Offset: 0

Hans G. Oberlack, Dec 09 2018

Sequence found by reading the line from 0, in the direction 0, 1, 12, ... in the triangle spiral.

			a(0) = 0
a(1) = a(1 - 1) + 1 = 0 + 1
a(2) = (3/2) * 2 * (2 + 2) = 3 * 4 = 12
a(3) = a(3 - 1) + 1 = 12 + 1 = 13
a(4) = (3/2) * 4*(4 + 2) = 3 * 2 * 6 = 6 * 6 = 36
a(5) = a(4) + 1 = 36 + 1 = 37.

Colin Barker, Table of n, a(n) for n = 0..1000
Hans G. Oberlack, Triangle spiral line 0-1-12-13
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A049598.

seq(coeff(series(-x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Dec 19 2018

a[0] = 0; a[n_] := a[n] = If[OddQ[n], a[n - 1] + 1, 3/2*n*(n + 2)]; Array[a, 50, 0] (* Amiram Eldar, Dec 09 2018 *)

concat(0, Vec(x*(1 + 11*x - x^2 + x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 09 2018

a(n) = (3/2)*n*(n+2) = A049598(n/2) if n even, a(n) = a(n-1)+1 if n odd.
G.f.: -x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 09 2018
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4. - Colin Barker, Dec 09 2018

Examples added by Hans G. Oberlack, Dec 20 2018

A274575 For m=1,2,3,... write all the 2^m binary vectors of length m in increasing order, and replace each vector with (number of 1's) - (number of 0's). Start with an initial 0 for the empty vector.

Original entry on oeis.org

0, -1, 1, -2, 0, 0, 2, -3, -1, -1, 1, -1, 1, 1, 3, -4, -2, -2, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 2, 2, 4, -5, -3, -3, -1, -3, -1, -1, 1, -3, -1, -1, 1, -1, 1, 1, 3, -3, -1, -1, 1, -1, 1, 1, 3, -1, 1, 1, 3, 1, 3, 3, 5, -6, -4, -4, -2, -4, -2, -2, 0, -4, -2, -2, 0, -2, 0, 0, 2, -4, -2, -2, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 2, 2, 4, -4, -2, -2, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 2, 2, 4, -2, 0, 0, 2, 0, 2, 2, 4, 0

Offset: 0

Hans G. Oberlack, Jun 28 2016

This is the sequence of To-And-Fro positions: Positions of all backward-forward combinations in lexicographical order when assigning -1 to a backward move and +1 to a forward move and starting at 0.

-a(n) are the slopes of the different segments, from left to right, of the successive steps in the construction of the Takagi (a.k.a. Blancmange) function. - Javier Múgica, Dec 31 2017

			Terms a(3) to a(6) correspond to the binary vectors 00, 01, 10, 11, which get replaced by -2, 0, 0, 2, respectively. Terms a(7) to a(14) correspond to the binary vectors 000, 001, ..., 111 which get replaced by -3, -1, ..., 3. a(0) = 0
a(1) = a('backward') = -1
a(2) = a('forward') = +1
a(3) = a('backward and backward') = -2
a(4) = a('backward and forward') = 0
a(5) = a('forward and backward') = 0
a(6) = a('forward and forward') = +2
a(7) = a('backward, backward and backward') = -3
a(8) = a('backward, backward and forward') = -1
Arranged as a tree read by rows:
               ______0______
              /             \
          __-1__           __1__
         /      \         /     \
       -2        0       0       2
       / \      / \     / \     / \
     -3  -1   -1   1  -1   1   1   3
. - _John Tyler Rascoe_, Sep 23 2023

John Tyler Rascoe, Table of n, a(n) for n = 0..8190

Cf. A037861.

Dim a(2*k+2)
a(0) = 0
For n = 0 To k
  a(2 * n + 1) = a(n) - 1
  a(2 * n + 2) = a(n) + 1
Next n

def A274575_list(nmax):
    A = [0]
    for n in range(0,nmax):
        A.append(A[n//2]-(-1)**n)
    return(A)
print(A274575_list(119)) # John Tyler Rascoe, Sep 23 2023

a(2*n + 1) = a(n) - 1; a(2*n + 2) = a(n) + 1.

Edited by N. J. A. Sloane, Jul 27 2016

A263997 Sequence of block lengths in a block spiral of width 1.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33

Offset: 1

Hans G. Oberlack, Oct 31 2015

A left-handed block spiral is created by the following pattern: Start with a 1 X 1 square [ a(1)=1 ]. Attach a block of length 2 and width 1 (a horizontal 1 X 2 rectangle) to the upper side of the square [ a(2)=2 ] pointing to the right. Attach to the corner a vertical 2 X 1 block (rectangle). This block has length 2, so a(3)=2. Continue with a horizontal 1 X 2 block, so a(4) = 2. And so on. See the sketch shown in the link.

Hans G. Oberlack, Sketch of a block spiral

Essentially the same as A008619.

a(n) = a(n-2) + 1 for n > 4.

O.g.f.: x*(1+x-x^2-x^3+x^4)/((1-x)*(1-x^2)). From A008619. - Wolfdieter Lang, Jan 05 2016

Edited by Wolfdieter Lang, Jan 05 2016

cp's OEIS Frontend

User: Hans G. Oberlack

A367914 Movement sequence in the counter-clockwise undulating spiral, whereby 1, 2, 3, 4 represent moves to the right, down, left and up.

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A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.

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A359216 X-coordinates of a point moving in a counterclockwise undulating spiral in a square grid.

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A359058 a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.

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A343203 Sum of x- and y-coordinate of a point moving in a triangular spiral.

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A322465 Numbers on the 0-9-10-line in a spiral on an equilateral triangular lattice.

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Maple

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A322462 Numbers on the 0-1-12 line in a spiral on a grid of equilateral triangles.

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Maple

Mathematica

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A274575 For m=1,2,3,... write all the 2^m binary vectors of length m in increasing order, and replace each vector with (number of 1's) - (number of 0's). Start with an initial 0 for the empty vector.

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