cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A054552 a(n) = 4*n^2 - 3*n + 1.

Original entry on oeis.org

1, 2, 11, 28, 53, 86, 127, 176, 233, 298, 371, 452, 541, 638, 743, 856, 977, 1106, 1243, 1388, 1541, 1702, 1871, 2048, 2233, 2426, 2627, 2836, 3053, 3278, 3511, 3752, 4001, 4258, 4523, 4796, 5077, 5366, 5663, 5968, 6281, 6602, 6931, 7268, 7613, 7966, 8327
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 09 2000

Keywords

Comments

Also indices in any square spiral organized like A054551.
Equals binomial transform of [1, 1, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 11 2008
Ulam's spiral (E spoke). - Robert G. Wilson v, Oct 31 2011
For n > 0: left edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012

Examples

			The spiral begins:
.
197-196-195-194-193-192-191-190-189-188-187-186-185-184-183
  |                                                       |
198 145-144-143-142-141-140-139-138-137-136-135-134-133 182
  |   |                                               |   |
199 146 101-100--99--98--97--96--95--94--93--92--91 132 181
  |   |   |                                       |   |   |
200 147 102  65--64--63--62--61--60--59--58--57  90 131 180
  |   |   |   |                               |   |   |   |
201 148 103  66  37--36--35--34--33--32--31  56  89 130 179
  |   |   |   |   |                       |   |   |   |   |
202 149 104  67  38  17--16--15--14--13  30  55  88 129 178
  |   |   |   |   |   |               |   |   |   |   |   |
203 150 105  68  39  18   5---4---3  12  29  54  87 128 177
  |   |   |   |   |   |   |       |   |   |   |   |   |   |
204 151 106  69  40  19   6   1---2  11  28  53  86 127 176
  |   |   |   |   |   |   |           |   |   |   |   |   |
205 152 107  70  41  20   7---8---9--10  27  52  85 126 175
  |   |   |   |   |   |                   |   |   |   |   |
206 153 108  71  42  21--22--23--24--25--26  51  84 125 174
  |   |   |   |   |                           |   |   |   |
207 154 109  72  43--44--45--46--47--48--49--50  83 124 173
  |   |   |   |                                   |   |   |
208 155 110  73--74--75--76--77--78--79--80--81--82 123 172
  |   |   |                                           |   |
209 156 111-112-113-114-115-116-117-118-119-120-121-122 171
  |   |                                                   |
210 157-158-159-160-161-162-163-164-165-166-167-168-169-170
  |
211-212-213-214-215-216-217-218-219-220-221-222-223-224-225
.
- _Robert G. Wilson v_, Jul 04 2014

Links

Crossrefs

Cf. A033293, A054551, A108781.
Spokes of square spiral: A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215.

Programs

Formula

G.f.: (1 - x + 8*x^2)/(1-x)^3.
a(n) = 8*n + a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 07 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=11. - Harvey P. Dale, Oct 10 2011
E.g.f.: exp(x)*(1 + x + 4*x^2). - Stefano Spezia, May 14 2021
a(n) = A003215(n-1) + A000290(n). - Leo Tavares, Jul 21 2022

A054553 Prime number spiral (clockwise, Northeast spoke).

Original entry on oeis.org

2, 5, 41, 127, 269, 467, 751, 1093, 1523, 2027, 2621, 3299, 4007, 4861, 5749, 6763, 7867, 9041, 10273, 11719, 13121, 14723, 16319, 18061, 19963, 21851, 23857, 26021, 28289, 30661, 33029, 35531, 38201, 40993, 43759, 46751, 49789, 52957, 56197
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Comments

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Northeast, clockwise.

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Northeast.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A000040, A054554, A054551, A054555, A054564, A054566, A054568, A054570, A053999.

Programs

  • Magma
    [NthPrime(4*n^2 - 10*n + 7): n in [1..40]]; // Vincenzo Librandi, Aug 29 2018
  • Mathematica
    Table[ Prime[4n^2 - 10n + 7], {n, 1, 40} ]

Formula

a(n) = A000040(A054554(n+1)). - R. J. Mathar, Aug 29 2018

Extensions

Edited by Robert G. Wilson v, Feb 25 2002

A053999 Prime number spiral (clockwise, Northwest spoke).

Original entry on oeis.org

2, 23, 97, 227, 419, 661, 1009, 1427, 1879, 2437, 3083, 3803, 4637, 5519, 6481, 7573, 8737, 9931, 11321, 12743, 14327, 15877, 17609, 19471, 21391, 23327, 25469, 27739, 30047, 32429, 34849, 37501, 40231, 42943, 45989, 49037, 52121, 55381, 58579
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Comments

Largest prime in n-th shell of prime spiral.
Shell 1 comprises the primes 3 5 7 11 13 17 19 23; 3 is lowest, 23 is highest.

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Northwest.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A016754, A054551, A054553, A054555, A054564, A054566, A054566, A054570.

Programs

  • Mathematica
    Table[ Prime[4n^2 + 4n + 1], {n, 0, 40} ]

Formula

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Northwest, clockwise.
a(n) = A000040(A016754(n)) . - R. J. Mathar, Aug 29 2018

Extensions

Edited by Robert G. Wilson v, Feb 25 2002

A054555 Prime number spiral (clockwise, East spoke).

Original entry on oeis.org

2, 7, 47, 139, 283, 503, 797, 1151, 1579, 2089, 2689, 3361, 4099, 4967, 5861, 6883, 8011, 9199, 10457, 11903, 13313, 14887, 16547, 18269, 20161, 22091, 24083, 26297, 28573, 30941, 33347, 35899, 38593, 41299, 44111, 47149, 50131, 53327, 56597
Offset: 0

Views

Author

Enoch Haga, and G. L. Honaker, Jr., Apr 10 2000

Keywords

Comments

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is East, clockwise.

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, East.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A054556, A054551, A054553, A054564, A054566, A054568, A054570, A053999.

Programs

  • Mathematica
    Table[ Prime[4n^2 - 9n + 6], {n, 1, 40} ]

Formula

a(n) = A000040(A054556(n+1)). - R. J. Mathar, Aug 29 2018

Extensions

Edited by Robert G. Wilson v, Feb 25 2002

A054566 Prime number spiral (clockwise, South spoke).

Original entry on oeis.org

2, 13, 67, 173, 347, 577, 877, 1249, 1697, 2243, 2833, 3541, 4289, 5179, 6131, 7159, 8293, 9473, 10799, 12251, 13709, 15289, 16987, 18749, 20681, 22619, 24671, 26839, 29129, 31541, 33911, 36559, 39217, 41981, 44839, 47903, 50989, 54163, 57347
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Comments

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is South, clockwise.

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, South.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A000040, A054567, A054551, A054553, A054555, A054564, A054568, A054570, A053999.

Programs

  • Mathematica
    Table[ Prime[4n^2 - 7n + 4], {n, 1, 40} ]

Formula

a(n) = A000040(A054567(n+1)). - Omar E. Pol, Feb 20 2022

Extensions

Edited by Robert G. Wilson v, Feb 25 2002

A054564 Prime number spiral (clockwise, Southeast spoke).

Original entry on oeis.org

2, 11, 59, 157, 313, 547, 829, 1201, 1621, 2153, 2749, 3463, 4217, 5059, 6011, 7001, 8167, 9343, 10631, 12071, 13513, 15107, 16759, 18481, 20399, 22343, 24371, 26591, 28807, 31231, 33617, 36229, 38891, 41647, 44501, 47533, 50549, 53759, 56957
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Comments

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Southeast, clockwise.

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Southeast.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A053755, A054551, A054553, A054555, A054566, A054568, A054570, A053999.

Programs

  • Mathematica
    Table[ Prime[4n^2 + 1], {n, 0, 40} ]

Formula

a(n) = A000040(A053755(n)). - R. J. Mathar, Aug 29 2018

Extensions

Edited by Frank Ellermann, Feb 24 2002

A054568 Prime number spiral (clockwise, Southwest spoke).

Original entry on oeis.org

2, 17, 73, 191, 367, 607, 919, 1297, 1753, 2309, 2909, 3623, 4421, 5303, 6257, 7307, 8447, 9643, 10979, 12421, 13883, 15467, 17203, 19037, 20921, 22853, 24953, 27091, 29389, 31859, 34267, 36847, 39551, 42307, 45233, 48311, 51407, 54547, 57781
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Southwest.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A054569, A054551, A054553, A054555, A054564, A054566, A054570, A053999.

Programs

  • Mathematica
    Table[ Prime[4n^2 - 6n + 3], {n, 1, 40} ]

Formula

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Southwest, clockwise.
a(n) = A000040(A054569(n+1)). - R. J. Mathar, Aug 29 2018

Extensions

Edited by Robert G. Wilson v, Feb 25 2002

A054570 Prime number spiral (clockwise, West spoke).

Original entry on oeis.org

2, 19, 83, 199, 389, 641, 967, 1361, 1823, 2377, 3001, 3709, 4517, 5419, 6353, 7477, 8623, 9791, 11159, 12577, 14083, 15667, 17417, 19273, 21149, 23063, 25229, 27431, 29683, 32183, 34543, 37171, 39877, 42641, 45599, 48673, 51719, 54973, 58171
Offset: 0

Views

Author

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

Keywords

Examples

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, West.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Links

Crossrefs

Cf. A033951, A054551, A054553, A054555, A054564, A054566, A054570, A053999.

Programs

  • Mathematica
    Table[ Prime[4n^2 + 3n + 1], {n, 0, 40} ]

Formula

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is West, clockwise.
a(n) = A000040(A033951(n)). - R. J. Mathar, Aug 29 2018

Extensions

Edited by Frank Ellermann, Feb 24 2002
Showing 1-8 of 8 results.

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