A214526 -id:A214526 - cp's OEIS frontend

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

1, 1, 1, 2, 3, 5, 8, 11, 21, 40, 47, 93, 180, 203, 397, 796, 1576, 1675, 3305, 6636, 13192, 14004, 27607, 55029, 110192, 220024, 226740, 450123, 898661, 1798700, 3594248, 3704800, 7354303, 14681369, 29349536, 58710640, 117394896, 119196748, 237492079

Offset: 0

Alec Jones and Peter Kagey, May 09 2020

nonn

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).

The central cell gets index 0 (and we fill it in with the value a(0)=1).

			Spiral begins:
     3----2----1
     |         |
     5    1----1   47
     |              |
     8---11---21---40
a(11) = 47 = 1 + 1 + 5 + 40, the sum of the cells in its row and column.

Peter Kagey, Table of n, a(n) for n = 0..2499

Cf. A280027.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334742.
x- and y-coordinates are given by A174344 and A274923, respectively.

\\ here P(n) returns A174344 and A274923 as pair.
P(n)={my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, [k, 3*k+n], [-k-n, k]), if(nAndrew Howroyd, May 09 2020

A225590 Primes p such that A217015(p) is a prime number. That is, when applying the walk of a square spiral to the data of rotated-square spiral, on step p a prime number is hit.

Original entry on oeis.org

2, 11, 17, 23, 53, 61, 67, 139, 149, 151, 163, 251, 263, 269, 281, 397, 421, 431, 541, 547, 557, 607, 619, 743, 773, 809, 1021, 1039, 1229, 1279, 1291, 1303, 1361, 1553, 1601, 1619, 1637, 1871, 1901, 1949, 2003, 2239, 2251, 2267, 2281, 2287, 2309, 2311, 2347, 2381, 2393

Offset: 1

Alex Ratushnyak, May 15 2013

nonn

Corresponding primes with prime indices, in sorted order: A225754.

Cf. A217015, A217296, A215468, A214526, A225754.

A225754 Primes p such that A217296(p) is a prime number. That is, when applying the walk of rotated-square spiral to the data of square spiral, on step p a prime number is hit.

Original entry on oeis.org

5, 11, 13, 29, 31, 41, 67, 71, 73, 79, 127, 137, 193, 199, 211, 293, 313, 421, 499, 503, 619, 631, 647, 661, 673, 773, 811, 967, 991, 1013, 1129, 1163, 1553, 1567, 1597, 1601, 1607, 1747, 1777, 1783, 1789, 1801, 1831, 1861, 1997, 2039, 2053, 2087, 2099, 2113, 2287, 2311

Offset: 1

Alex Ratushnyak, May 15 2013

nonn

Corresponding primes with prime indices, in sorted order: A225590. The intersection of a(n) with A225590 begins: 11, 67, 421, 619, 773, 1553, 1601, 2287, 2311, 2381, 2609, 3169, 3181, 3491, 3511, 4157, 4597, 4639, 6263, 7129, 7177, 7193, 8291, 9277.

Cf. A217015, A217296, A215468, A214526, A225590.

A235913 a(n) is the Manhattan distance between n^3 and (n+1)^3 in a square spiral of positive integers with 1 at the center.

Original entry on oeis.org

1, 3, 11, 15, 13, 9, 5, 21, 33, 59, 71, 49, 47, 35, 15, 13, 43, 73, 109, 123, 117, 109, 167, 141, 113, 77, 43, 5, 51, 95, 145, 201, 263, 281, 397, 413, 317, 333, 269, 239, 183, 121, 63, 11, 81, 147, 219, 307, 379, 471, 567, 623, 517, 569, 683, 503, 545, 473, 395, 311

Offset: 1

Alex Ratushnyak, Jan 16 2014

nonn

Spiral begins:
.
  49  26--27--28--29--30--31
   |   |                   |
  48  25  10--11--12--13  32
   |   |   |           |   |
  47  24   9   2---3  14  33
   |   |   |   |   |   |   |
  46  23   8   1   4  15  34
   |   |   |       |   |   |
  45  22   7---6---5  16  35
   |   |               |   |
  44  21--20--19--18--17  36
   |                       |
  43--42--41--40--39--38--37

			Manhattan distance between 2^3=8 and 3^3=27 is 3 in a square spiral, so a(2)=3.

Cf. A214526, A232113, A232114, A232115.

import math
def get_x_y(n):
  sr = int(math.sqrt(n-1))  # Ok for small n's
  sr = sr-1+(sr&1)
  rm = n-sr*sr
  d = (sr+1)//2
  if rm<=sr+1:
     return -d+rm, d
  if rm<=sr*2+2:
     return d, d-(rm-(sr+1))
  if rm<=sr*3+3:
     return d-(rm-(sr*2+2)), -d
  return -d, -d+rm-(sr*3+3)
for n in range(1, 77):
  x0, y0 = get_x_y(n**3)
  x1, y1 = get_x_y((n+1)**3)
  print(abs(x1-x0)+abs(y1-y0), end=', ')

A277214 a(n) is the Manhattan distance between n and 1 in a 3-dimensional cubic spiral of positive integers with 1..8 at the center (illustration in the comments).

Original entry on oeis.org

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

Offset: 1

Alex Ratushnyak, Oct 05 2016

nonn

Similar to A214526, but three-dimensional, and the core is 2 X 2 X 2 rather than 1 X 1.
The spiral begins as follows:
Level z=-2:
94  93  92  91  90
77  76  75  74  89
78  67  66  73  88
79  68  65  72  87
80  69  70  71  86
81  82  83  84  85
z=-1:
115 114 113 112 111
52  51  50  49 110
53  62  61  60 109
54  63  64  59 108
55  56  57  58 107
102 103 104 105 106
z=0:
136 135 134 133 132
39  38  37  48 131
40   3   2  47 130
41   4   1  46 129
42  43  44  45 128
123 124 125 126 127
z=1:
145 146 147 148 149
34  35  36  25 150
33   6   7  26 151
32   5   8  27 152
31  30  29  28 153
158 157 156 155 154
z=2:
166 167 168 169 170
21  22  23  24 171
20  11  12  13 172
19  10   9  14 173
18  17  16  15 174
179 178 177 176 175
z=3:
187 188 189 190 191
204 205 206 207 192
203 214 215 208 193
202 213 216 209 194
201 212 211 210 195
200 199 198 197 196
Algorithm sketch:
At every x-y plane the direction is clockwise if z > 0 and counterclockwise if z <= 0.
After an N*N cube is complete and we start building an M*M cube, M=N+2:
2a. The spiral at the first new edge of the M*M cube progresses from center to edges, in the same way as the A214526 spiral, e.g., z=-2 in the illustration.
2b. Between the first and last z-edges the spiral progresses according to item 1.
2c. The spiral at the last new edge of the M*M cube progresses from edges to center, e.g., z=3 in the illustration.

Cf. A214526.

abs( a(n) - a(n-1) ) = 1.

A368643 Manhattan distance from the origin of triangular number point T(n) = n*(n+1)/2 in a square spiral with point 0 at the origin.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 6, 3, 7, 6, 6, 9, 5, 10, 8, 9, 11, 8, 14, 9, 13, 12, 12, 17, 9, 18, 14, 15, 19, 12, 22, 15, 19, 20, 16, 25, 15, 24, 20, 21, 27, 16, 30, 21, 25, 28, 20, 35, 21, 30, 28, 25, 35, 20, 36, 27, 31, 36, 24, 43, 27, 36, 36, 29, 45, 26, 42, 35, 35, 44, 28, 49, 33, 42, 44, 33, 55, 32, 48, 43, 39, 54, 30

Offset: 0

Tamas Sandor Nagy, Jan 01 2024

			Showing the Manhattan distance for a(4), that is, from the origin to the fourth triangular number T(4) = 10, as being 2:
  +----15-----+-----+-----+     +
  |                       |     |
  |                       |     |
  +     +-----3-----+     +    28
  |     |           |     |     |
  |     |           |     |     |
  +     +     0-----1    10     +
  |     |     ^---a(4)=2--^     |
  |     |                 |     |
  +     6-----+-----+-----+     +
  |                             |
  |                             |
  +----21-----+-----+-----+-----+

Cf. A214526, A000217.

a(n) = A214526(n*(n+1)/2 + 1).

cp's OEIS Frontend

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

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A225590 Primes p such that A217015(p) is a prime number. That is, when applying the walk of a square spiral to the data of rotated-square spiral, on step p a prime number is hit.

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A225754 Primes p such that A217296(p) is a prime number. That is, when applying the walk of rotated-square spiral to the data of square spiral, on step p a prime number is hit.

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A235913 a(n) is the Manhattan distance between n^3 and (n+1)^3 in a square spiral of positive integers with 1 at the center.

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Python

A277214 a(n) is the Manhattan distance between n and 1 in a 3-dimensional cubic spiral of positive integers with 1..8 at the center (illustration in the comments).

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A368643 Manhattan distance from the origin of triangular number point T(n) = n*(n+1)/2 in a square spiral with point 0 at the origin.

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