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.

A054556 a(n) = 4*n^2 - 9*n + 6.

Original entry on oeis.org

1, 4, 15, 34, 61, 96, 139, 190, 249, 316, 391, 474, 565, 664, 771, 886, 1009, 1140, 1279, 1426, 1581, 1744, 1915, 2094, 2281, 2476, 2679, 2890, 3109, 3336, 3571, 3814, 4065, 4324, 4591, 4866, 5149, 5440, 5739, 6046, 6361, 6684, 7015, 7354, 7701, 8056, 8419, 8790
Offset: 1

Views

Author

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

Keywords

Comments

Move in 1-4 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 3, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
Ulam's spiral (N spoke). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonpositive m. - Bruno Berselli, Jan 06 2016

Links

Crossrefs

Cf. A054555, A068225, A054552, A054554, A054567, A054569, A033951.
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.
Cf. A001105, A058331, A130883.

Programs

Formula

a(n)^2 = Sum_{i = 0..2*(4*n-5)} (4*n^2-13*n+9+i)^2*(-1)^i = ((n-1)*(4*n-5)+1)^2. - Bruno Berselli, Apr 29 2010
From Harvey P. Dale, Aug 21 2011: (Start)
a(0)=1, a(1)=4, a(2)=15; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(6*x^2+x+1)/(x-1)^3. (End)
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n - 2, 2) + 2*(n - 1)^2 + 1.
a(n) = A000384(n-1) + A058331(n-1).
a(n) = A130883(n-1) + A001105(n-1). (End)
E.g.f.: exp(x)*(6 - 5*x + 4*x^2) - 6. - Stefano Spezia, Apr 24 2024

Extensions

Edited by Frank Ellermann, Feb 24 2002
Incorrect formula deleted by N. J. A. Sloane, Aug 02 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

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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