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 1-4 of 4 results.

A007389 7th-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 2, 13, 2, 15, 2, 17, 11, 19, 22, 21, 35, 23, 50, 25, 67, 36, 86, 58, 107, 93, 130, 143, 155, 210, 191, 296, 249, 403, 342, 533, 485, 688, 695, 879, 991, 1128, 1394, 1470, 1927, 1955, 2615, 2650, 3494, 3641, 4622, 5035, 6092, 6962, 8047
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. Yanco and A. Bagchi, K-th order maximal independent sets in path and cycle graphs, J. Graph Theory, submitted, 1994, apparently unpublished.

Links

Crossrefs

Cf. A001608, A007387, A007388.

Formula

Empirical g.f.: x^2*(7*x^14 + 5*x^12 + 3*x^10 - 2*x^7 - 2*x^5 - 2*x^3 - 3*x - 2) / (x^9 + x^2 - 1). - Colin Barker, Mar 29 2014
Theorem: a(n) = Sum_{j=0..floor((n-g)/(2*g))} (2*n/(n-2*(g-2)*j-(g-2))) * Hypergeometric2F1([-(n-2g*j-g)/2,-(2j+1)], [1], 1), g = 9, n >= g and n an odd integer. - Richard Turk, Oct 14 2019 For proof see attached text file.

A133394 a(n)=a(n-2)+a(n-5).

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 7, 2, 9, 7, 11, 14, 13, 23, 20, 34, 34, 47, 57, 67, 91, 101, 138, 158, 205, 249, 306, 387, 464, 592, 713, 898, 1100, 1362, 1692, 2075, 2590, 3175, 3952, 4867, 6027, 7457, 9202, 11409, 14069, 17436, 21526, 26638, 32935, 40707, 50371, 62233
Offset: 1

Views

Author

G. Reed Jameson (Reedjameson(AT)yahoo.com), Nov 23 2007

Keywords

Comments

Perrin-like prime-divisibility sequence, but based upon template 7=5+2 in place of 5=3+2.
1. Apparently identical to A007387 but for latter's third term 3. 2. Attention directed to remainder upon division of a term by its (composite) argument, when latter =1 or 5 (mod 6). Possible factorization tool for impostor candidate primes? 3. Recurrence period, any length-five string of term values (mod 6) found in the sequence: 2^3*13*31, to Perrin's three-term period of 7*13. Note 13= 2*6+1, 31 = 5*6+1. 4. Query: Smallest pseudoprime >9. 5. Query: Closed form for n-th term.
Semiprimes a= 9, 14, 34, 57, 91 etc. are at the indices n=9, 12, 16, 17, 19, 21, 24, 25, 26, 31, 32, 40, 44, 45, 51, 53, 59, 66, 72, 76, 80, 110 etc. - R. J. Mathar, Nov 24 2007

Links

Crossrefs

Cf. A007387, A001608, A135435, A136598.

Programs

  • Mathematica
    LinearRecurrence[{0,1,0,0,1},{0,2,0,2,5},60] (* Harvey P. Dale, Oct 21 2015 *)
  • PARI
    {a(n) = if( n<0, n = 1 - n; polsym(x^5 + x^2 - 1, n)[n], n++; polsym(x^5 - x^3 - 1, n)[n])} /* Michael Somos, Feb 12 2012 */

Formula

O.g.f.: -x*(2+5*x^3)/(-1+x^2+x^5). - R. J. Mathar, Nov 24 2007
Rewritten, Mathar's o.g.f. resembles a logarithmic derivative: -(5*x^4 + 2*x) / (x^5 +x^2-1). Any significance? - G. Reed Jameson (Reedjameson(AT)yahoo.com), Dec 13 2007, Dec 16 2007
a(-n) = A136598(n).

Extensions

More terms from R. J. Mathar, Nov 24 2007

A007388 5th-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 2, 13, 9, 15, 18, 17, 29, 19, 42, 28, 57, 46, 74, 75, 93, 117, 121, 174, 167, 248, 242, 341, 359, 462, 533, 629, 781, 871, 1122, 1230, 1584, 1763, 2213, 2544, 3084, 3666, 4314, 5250, 6077, 7463, 8621, 10547
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. Yanco and A. Bagchi, K-th order maximal independent sets in path and cycle graphs, J. Graph Theory, submitted, 1994, apparently unpublished.

Links

Crossrefs

Cf. A001608, A007387, A007389.

Formula

Empirical g.f.: x^2*(5*x^10+3*x^8-2*x^5-2*x^3-3*x-2) / (x^7+x^2-1). - Colin Barker, Mar 29 2014
For n >= 13: a(n) = a(n-2) + a(n-7). - Sean A. Irvine, Jan 02 2018
a(n) = Sum_{j=0..floor((n-g)/(2*g))} (2*n/(n-2*(g-2)*j-(g-2))) * Hypergeometric2F1([-(n-2g*j-g)/2,-(2j+1)], [1], 1), g = 7, n >= g and n an odd integer. - Richard Turk, Oct 14 2019

Extensions

Typo in data (242 was inadvertently repeated) fixed by Colin Barker, Mar 29 2014
More terms from Sean A. Irvine, Jan 02 2018

A007392 Number of strict 3rd-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 0, 21, 5, 32, 17, 45, 38, 65, 70, 99, 115, 156, 180, 247, 279, 385, 435, 590, 682, 896, 1067, 1360, 1657, 2073, 2553, 3173, 3913, 4865, 5986, 7455, 9159, 11407, 14024, 17434, 21479, 26636, 32886, 40705, 50320
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. Yanco and A. Bagchi, "K-th order maximal independent sets in path and cycle graphs", Journal of Graph Theory, submitted, 1994, apparently unpublished.

Links

Crossrefs

Cf. A007387.

Formula

Conjecture: a(n) = 3*a(n-2) - 3*a(n-4) + a(n-5) + a(n-6) - 2*a(n-7) + a(n-9) with g.f. x^10*(-5+3*x^2)/((x^5+x^2-1)*(x-1)^2*(1+x)^2). - R. J. Mathar, Oct 30 2009
a(n) = A007387(n) - b(n) where b(1) = 0, b(2*n+1) = 2*n+1, b(2*n) = 2. - Sean A. Irvine, Jan 02 2018

Extensions

More terms from Sean A. Irvine, Jan 02 2018
Showing 1-4 of 4 results.

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

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