A133394 -id:A133394 - cp's OEIS frontend

A007387 Number of 3rd-order maximal independent sets in cycle graph.

0, 2, 3, 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

N. J. A. Sloane, Mira Bernstein

nonn

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.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. Yanco, Letter and Email to N. J. A. Sloane, 1994
R. Yanco and A. Bagchi, K-th order maximal independent sets in path and cycle graphs, Unpublished manuscript, 1994. (Annotated scanned copy)

Cf. A001608, A007388, A007389.

R:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x^2*(2+3*x+2*x^3-3*x^6)/(1-x^2-x^5) )); // G. C. Greubel, Oct 19 2019

seq(coeff(series(x^2*(2+3*x+2*x^3-3*x^6)/(1-x^2-x^5), x, n+1), x, n), n = 1..50); # G. C. Greubel, Oct 19 2019

Rest[CoefficientList[Series[x^2*(2+3*x+2*x^3-3*x^6)/(1-x^2-x^5), {x, 0, 50}], x]] (* Harvey P. Dale, Oct 23 2011 *)

my(x='x+O('x^50)); concat([0], Vec(x^2*(2+3*x+2*x^3-3*x^6)/(1-x^2-x^5))) \\ G. C. Greubel, Oct 19 2019

def A007387_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(2+3*x+2*x^3-3*x^6)/(1-x^2-x^5)).list()
a=A007387_list(50); a[1:] # G. C. Greubel, Oct 19 2019

For n >= 9: a(n) = a(n-2) + a(n-5) per A133394. - G. Reed Jameson (Reedjameson(AT)yahoo.com), Dec 13 2007, Dec 16 2007
G.f.: x^2*(2 + 3*x + 2*x^3 - 3*x^6)/(1 - x^2 - x^5). - R. J. Mathar, Oct 30 2009
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), with g = 5, n >= g, and n an odd integer. - Richard Turk, Oct 14 2019

More terms from Harvey P. Dale, Oct 23 2011

A135435 a(n) = a(n-4) + a(n-7) with a(0), ..., a(6) = [7,0,0,0,4,0,0].

Original entry on oeis.org

7, 0, 0, 0, 4, 0, 0, 7, 4, 0, 0, 11, 4, 0, 7, 15, 4, 0, 18, 19, 4, 7, 33, 23, 4, 25, 52, 27, 11, 58, 75, 31, 36, 110, 102, 42, 94, 185, 133, 78, 204, 287, 175, 172, 389, 420, 253, 376, 676, 595, 425, 765, 1096, 848, 801, 1441, 1691, 1273, 1566, 2537, 2539, 2074, 3007, 4228

Offset: 0

G. Reed Jameson (Reedjameson(AT)yahoo.com), Dec 13 2007, Dec 16 2007

Of interest because {7,11} is the earliest pair of typical primes belonging to a single hexad. Herein "pseudoprime" means sequence-specific psp. (i.e. dividing its term with rem. 0), not general number-theoretic psp. The only psp.s of concern, from the standpoint of primality testing, being those congruent to 1 or 5 (mod 6), are the six quadratics of the present zero-termed primes 5,13,17 the only relevant psp.s of this sequence? Or are there additional examples > 289?

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 0, 0, 1).

Cf. A133394.

LinearRecurrence[{0,0,0,1,0,0,1},{7,0,0,0,4,0,0},70] (* Harvey P. Dale, Jan 20 2013 *)

Vec((7-3*x^4)/(1-x^4-x^7) + O(x^80)) \\ Michel Marcus, Oct 14 2016

G.f.: (7-3*x^4)/(1-x^4-x^7). - R. J. Mathar, Oct 24 2009

More terms from R. J. Mathar, Oct 24 2009

A136598 Expansion of g.f. (2*x^3 + 5) / ( -x^5 + x^3 + 1).

Original entry on oeis.org

5, 0, 0, -3, 0, 5, 3, 0, -8, -3, 5, 11, 3, -13, -14, 2, 24, 17, -15, -38, -15, 39, 55, 0, -77, -70, 39, 132, 70, -116, -202, -31, 248, 272, -85, -450, -303, 333, 722, 218, -783, -1025, 115, 1505, 1243, -898, -2530, -1128, 2403, 3773, 230, -4933, -4901, 2173, 8706, 5131, -7106, -13607

Offset: 0

Michael Somos, Jan 11 2008

			5 - 3*x^3 + 5*x^5 + 3*x^6 - 8*x^8 - 3*x^9 + 5*x^10 + 11*x^11 + 3*x^12 - ...

Index entries for linear recurrences with constant coefficients, signature (0,0,-1,0,1).

Cf. A133394.

{a(n) = if( n<0, n = 1 - n; polsym(x^5 - x^3 - 1, n)[n], n++; polsym(x^5 + x^2 - 1, n)[n])}

a(n) = -a(n-3) + a(n-5). a(-n) = A133394(n).

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A007387 Number of 3rd-order maximal independent sets in cycle graph.

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A135435 a(n) = a(n-4) + a(n-7) with a(0), ..., a(6) = [7,0,0,0,4,0,0].

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A136598 Expansion of g.f. (2*x^3 + 5) / ( -x^5 + x^3 + 1).

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