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.

A054567 a(n) = 4*n^2 - 7*n + 4.

Original entry on oeis.org

1, 6, 19, 40, 69, 106, 151, 204, 265, 334, 411, 496, 589, 690, 799, 916, 1041, 1174, 1315, 1464, 1621, 1786, 1959, 2140, 2329, 2526, 2731, 2944, 3165, 3394, 3631, 3876, 4129, 4390, 4659, 4936, 5221, 5514, 5815, 6124, 6441, 6766, 7099, 7440, 7789, 8146, 8511, 8884
Offset: 1

Views

Author

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

Keywords

Comments

The number 1 is placed in the middle of a sheet of squared paper and the numbers 2, 3, 4, 5, 6, etc. are written in a clockwise spiral around 1, as in A068225 etc. This sequence is read off along one of the rays from 1.
Ulam's spiral (W spoke of A054552). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonnegative m. - Bruno Berselli, Jan 06 2016
The sequence forms the 1x2 diagonal of the square maze arrangement in A081344. - Jarrod G. Sage, Jul 17 2024

Links

Crossrefs

Cf. A054566, A068225, A054552, A054554, A054556, A054569, A033951, A204674.
Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...
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.

Programs

Formula

a(n) = 8*n+a(n-1)-11 for n>1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
a(n) = A204674(n-1) / n. - Reinhard Zumkeller, Jan 18 2012
From Colin Barker, Oct 25 2014: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
G.f.: -x*(4*x^2+3*x+1) / (x-1)^3. (End)
E.g.f.: exp(x)*(4 - 3*x + 4*x^2) - 4. - Stefano Spezia, Apr 24 2024
a(n) = A016742(n-1) + n. - Jarrod G. Sage, Jul 17 2024

Extensions

Edited by Frank Ellermann, Feb 24 2002
Typo fixed by Charles R Greathouse IV, Oct 28 2009

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

A054551 Prime number spiral (clockwise, North spoke).

Original entry on oeis.org

2, 3, 31, 107, 241, 443, 709, 1049, 1471, 1973, 2539, 3191, 3911, 4729, 5651, 6637, 7699, 8867, 10133, 11503, 12941, 14537, 16073, 17863, 19727, 21601, 23609, 25759, 27967, 30319, 32719, 35201, 37831, 40627, 43391, 46399, 49411, 52553, 55813
Offset: 0

Views

Author

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

Keywords

Comments

Smallest prime in n-th shell of prime spiral.
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is North, clockwise.
Shell 1 comprises the primes 3 5 7 11 13 17 19 23; 3 is lowest, 23 is highest.
The wheel may be rotated, but the sequences though pointing in different directions, will remain the same.

Examples

			Begin a prime number spiral at zero, proceed clockwise, North.
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, A054552, A054553, A054555, A054564, A054566, A054568, A054570, A053999.

Programs

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

Formula

a(n) = A000040(A054552(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

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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