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 1-7 of 7 results.

A054569 a(n) = 4*n^2 - 6*n + 3.

Original entry on oeis.org

1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163, 2353, 2551, 2757, 2971, 3193, 3423, 3661, 3907, 4161, 4423, 4693, 4971, 5257, 5551, 5853, 6163, 6481, 6807, 7141, 7483, 7833, 8191
Offset: 1

Views

Author

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

Keywords

Comments

Move in 1-7 direction in a spiral organized like A068225 etc.
Third row of A082039. - Paul Barry, Apr 02 2003
Inverse binomial transform of A036826. - Paul Barry, Jun 11 2003
Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - Johannes W. Meijer, May 20 2011
Ulam's spiral (SW spoke). - Robert G. Wilson v, Oct 31 2011

Links

Crossrefs

Cf. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.
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+1) = 4*n^2 + 2*n + 1. - Paul Barry, Apr 02 2003
a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - Paul Barry, Jun 11 2003
Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Mar 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1+x)*(1+3*x)/(1-x)^3. (End)
a(n) = A000384(n) + A000384(n-1). - Bruce J. Nicholson, May 07 2017
E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - G. C. Greubel, Jul 04 2019
Sum_{n>=1} 1/a(n) = A339237. - R. J. Mathar, Jan 22 2021

Extensions

Edited by Frank Ellermann, Feb 24 2002

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

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

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

A122566 a(n) = prime(n^2 + n + 1).

Original entry on oeis.org

2, 5, 17, 41, 73, 127, 191, 269, 367, 467, 607, 751, 919, 1093, 1297, 1523, 1753, 2027, 2309, 2621, 2909, 3299, 3623, 4007, 4421, 4861, 5303, 5749, 6257, 6763, 7307, 7867, 8447, 9041, 9643, 10273, 10979, 11719, 12421, 13121, 13883, 14723, 15467, 16319
Offset: 0

Views

Author

Miklos Kristof, Sep 21 2006

Keywords

Comments

Union of A054553 and A054568. Primes appearing on the northeast and southwest spokes of the (clockwise) prime number spiral. - Vincenzo Librandi, Nov 06 2024

Examples

			a(2) = 17 = P(2^2+2+1) = P(7).

Links

Crossrefs

Even bisection gives A054568; odd bisection gives A054553(n>0).
Cf. A000040.

Programs

  • Magma
    [NthPrime(n^2+n+1): n in [0..60]]; // G. C. Greubel, Oct 29 2024
  • Maple
    seq(ithprime(k^2+k+1),k=0..60);
  • Mathematica
    Table[Prime[k^2+k+1],{k,0,50}] (* Harvey P. Dale, Apr 01 2013 *)
  • SageMath
    [nth_prime(n^2+n+1) for n in range(61)] # G. C. Greubel, Oct 29 2024

Formula

a(n) = prime(n^2+n+1). - Wesley Ivan Hurt, Nov 28 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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