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.

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A007382 Number of strict (-1)st-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 3, 4, 11, 16, 32, 49, 87, 137, 231, 369, 608, 978, 1595, 2574, 4179, 6754, 10944, 17699, 28655, 46355, 75023, 121379, 196416, 317796, 514227, 832024, 1346267
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.

Links

Crossrefs

Equals A054451(n+1) - 1.

Programs

  • Mathematica
    Table[Sum[Binomial[n - i + 1, i], {i, Floor[(n - 1)/2]}], {n, 30}] (* or *)
Rest@ Abs@ CoefficientList[Series[x^3*(x^3 + 2 x^2 - x - 3)/((1 - x - x^2) (1 - x^2)^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 19 2017 *)

Formula

John W. Layman observes that if b(n) = 1+A007382(n) then b(n) = b(n-1) + 3b(n-2) - 2b(n-3) - 3b(n-4) + b(n-5) + b(n-6) for all 27 terms shown.
G.f.: x^3*(x^3+2x^2-x-3)/((1-x-x^2)*(1-x^2)^2).
a(n) = Sum_{i=1..floor((n-1)/2)} C(n-i+1, i). - Wesley Ivan Hurt, Sep 19 2017

A007381 7th-order maximal independent sets in path graph.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 4, 7, 7, 8, 11, 9, 16, 11, 22, 15, 29, 22, 37, 33, 46, 49, 57, 71, 72, 100, 94, 137, 127, 183, 176, 240, 247, 312, 347, 406, 484, 533, 667, 709, 907, 956, 1219, 1303, 1625, 1787, 2158, 2454, 2867, 3361, 3823, 4580
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Examples

			G.f. = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 4*x^6 + 5*x^7 + 2*x^8 + 6*x^9 + ...

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.

Links

Formula

Empirical g.f.: -x*(x^8+x^7+x^5+x^3+2*x+1) / (x^9+x^2-1). - Colin Barker, Mar 29 2014
a(n) = T(2, 9, n + 9) where T(a, b, n) = Sum_{a*x+b*y = n, x >= 0, y >= 0} binomial(x+y, y). - Sean A. Irvine, Jan 02 2018

Extensions

a(22) corrected by Colin Barker, Mar 29 2014
More terms from Sean A. Irvine, Jan 02 2018

A007383 Number of strict first-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 6, 4, 11, 10, 20, 21, 36, 41, 64, 77, 113, 141, 199, 254, 350, 453, 615, 803, 1080, 1418, 1896, 2498, 3328, 4394, 5841, 7722, 10251, 13563, 17990, 23814, 31571, 41804, 55404, 73375, 97228, 128779, 170624, 226007, 299425
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.

Links

Crossrefs

Cf. A000931.

Formula

Empirical g.f.: -x^3 / ((x-1)^2*(x+1)^2*(x^3+x^2-1)). - Colin Barker, Mar 29 2014
a(n) = A000931(n + 6) - b(n) where b(2*n+1) = 1 and b(2*n) = n+1. - Sean A. Irvine, Jan 02 2018

Extensions

More terms from Sean A. Irvine, Jan 02 2018

A007384 Number of strict 3rd-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 6, 1, 10, 4, 15, 10, 22, 20, 33, 35, 51, 57, 80, 90, 125, 141, 193, 221, 295, 346, 449, 539, 684, 834, 1045, 1283, 1600, 1967, 2451, 3012, 3752, 4612, 5738, 7063, 8770, 10815, 13403, 16553, 20488, 25323, 31326, 38726
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.

Links

Crossrefs

Cf. A001687.

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^5/((x^5+x^2-1)*(x-1)^2*(1+x)^2). [From R. J. Mathar, Oct 30 2009]
a(n) = A001687(n + 6) - b(n) where b(2*n+1) = 1 and b(2*n) = n+1. - Sean A. Irvine, Jan 02 2018

Extensions

More terms from Sean A. Irvine, Jan 02 2018

A007385 Number of strict 5th-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 6, 0, 10, 1, 15, 4, 21, 10, 28, 20, 37, 35, 50, 56, 70, 84, 101, 121, 148, 171, 217, 241, 315, 342, 451, 490, 638, 707, 896, 1022, 1256, 1473, 1765, 2111, 2492, 3007, 3535, 4263, 5030, 6028, 7164, 8520, 10195
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. A007380.

Formula

Apparently a(n)= 3*a(n-2) -3*a(n-4) +a(n-6) +a(n-7) -2*a(n-9) +a(n-11) with g.f. -x^7/((x^7+x^2-1)*(x-1)^2*(1+x)^2). [From R. J. Mathar, Oct 30 2009]
a(n) = A007380(n) - b(n) where b(2*n+1) = 1 and b(2*n) = n+1.- Sean A. Irvine, Jan 02 2018

Extensions

More terms from Sean A. Irvine, Jan 02 2018

A007386 Number of strict 7th-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 6, 0, 10, 0, 15, 1, 21, 4, 28, 10, 36, 20, 45, 35, 56, 56, 71, 84, 93, 120, 126, 165, 175, 221, 246, 292, 346, 385, 483, 511, 666, 686, 906, 932, 1218, 1278, 1624, 1761, 2157, 2427, 2866, 3333, 3822, 4551
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. A007381.

Formula

Apparently, g.f. = -x^9/((x^9+x^2-1)*(x-1)^2*(1+x)^2) with recurrence a(n)= 3*a(n-2) - 3*a(n-4) + a(n-6) + a(n-9) - 2*a(n-11) + a(n-13). - R. J. Mathar, Oct 30 2009
a(n) = A007381(n) - b(n) where b(2*n+1) = 1 and b(2*n) = n+1. - Sean A. Irvine, Jan 02 2018

Extensions

More terms from Sean A. Irvine, Jan 02 2018

A007390 Number of strict (-1)st-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 0, 0, 4, 5, 15, 21, 44, 66, 120, 187, 319, 507, 840, 1348, 2204, 3553, 5775, 9329, 15124, 24454, 39600, 64055, 103679, 167735, 271440, 439176, 710644, 1149821, 1860495, 3010317, 4870844, 7881162, 12752040, 20633203, 33385279, 54018483
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.

Links

Formula

a(n) = A000204(n) - b(n) where b(1) = 1, b(2*n+1) = 2*n+2, b(2*n) = 3. - Sean A. Irvine, Jan 02 2018
Conjectures from Colin Barker, Jun 14 2019: (Start)
G.f.: x^4*(4 + x - 2*x^2 - x^3) / ((1 - x)^2*(1 + x)^2*(1 - x - x^2)).
a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 3*a(n-4) + a(n-5) + a(n-6) for n>7.
(End)

Extensions

a(18) corrected and more terms from Sean A. Irvine, Jan 02 2018

A007391 Number of strict first-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 8, 3, 15, 11, 27, 26, 49, 53, 88, 102, 156, 190, 275, 346, 484, 621, 851, 1105, 1495, 1956, 2625, 3451, 4608, 6076, 8088, 10684, 14195, 18772, 24912, 32967, 43719, 57879, 76723, 101598, 134641, 178321, 236280, 312962, 414644
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.

Links

Crossrefs

Cf. A001608.

Formula

Empirical g.f.: x^6*(x^2-3) / ((x-1)^2*(x+1)^2*(x^3+x^2-1)). - Colin Barker, Mar 29 2014
a(n) = A001608(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

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

A007393 Number of strict 5th-order maximal independent sets in cycle graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 16, 0, 27, 0, 40, 7, 55, 23, 72, 50, 91, 90, 119, 145, 165, 217, 240, 308, 357, 427, 531, 592, 779, 832, 1120, 1189, 1582, 1720, 2211, 2499, 3082, 3619, 4312, 5201, 6075, 7412, 8619, 10494, 12285
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.

Links

Crossrefs

Cf. A007388.

Formula

Apparent g.f.: x^14*(-7+5*x^2)/((x^7+x^2-1)*(x-1)^2*(1+x)^2). [From R. J. Mathar, Oct 30 2009]
a(n) = A007388(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
Previous Showing 11-20 of 21 results. Next

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

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