A001998 -id:A001998 - cp's OEIS frontend

A056332 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of three different colors.

1, 1, 3, 8, 24, 65, 195, 564, 1677, 4976, 14883, 44452, 133224, 399113, 1196808, 3588840, 10764960, 32289855, 96864963, 290580040, 871725426, 2615132465, 7845353475, 23535926760, 70607649816, 211822550576

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A001998, A056314.

a1998[n_] := If[OddQ[n], (1/4)*(3^n + 4*3^((n-1)/2) + 1), (1/4)*(3^n + 2*3^(n/2) + 1)];
a[n_] := DivisorSum[n, MoebiusMu[#] a1998[n/#-1]&];
Array[a, 26] (* Jean-François Alcover, Jun 29 2018 *)

a(n) = Sum mu(d)*A001998(n/d-1) where d|n.

A038766 Triangle giving number of unbranched catapolytetragons, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 3, 9, 7, 4, 1, 5, 16, 29, 23, 10, 1, 5, 27, 62, 99, 69, 25, 1, 7, 39, 132, 275, 351, 229, 70, 1, 7, 55, 221, 643, 1121, 1249, 731, 196, 1, 9, 72, 367, 1278, 2997, 4584, 4437, 2385, 574, 1, 9, 93, 540, 2322, 6678, 13458, 18012, 15597, 7657, 1681

Offset: 0

N. J. A. Sloane, May 03 2000

			1; 1,1; 1,1,1; 1,3,3,2; 1,3,9,7,4; ...

S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Last diagonal is A001998.

LCM's of all cycles: A060113.

A323942 Irregular triangle read by rows giving the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 4, 7, 9, 3, 1, 10, 23, 29, 16, 5, 1, 25, 69, 99, 62, 27, 5, 1, 70, 229, 351, 275, 132, 39, 7, 1, 196, 731, 1249, 1121, 643, 221, 55, 7, 1, 574, 2385, 4437, 4584, 2997, 1278, 367, 72, 9, 1, 1681, 7657, 15597, 18012, 13458, 6678, 2322, 540, 93, 9, 1

Offset: 2

N. J. A. Sloane, Feb 09 2019

From Petros Hadjicostas, May 26 2019: (Start)
Let I(r, k) be the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes, which are generated from catafusenes by converting k of its r hexagons to tetragons. According to Cyvin et al. (1996), for r >= k, we have I(r, k) = 1/4 *(binomial(r, k) + (r - 2)! * (r^2 + (4 * k - 1) * r + 4 * k * (k - 2)) * 3^(r - k - 2)/(k! * (r - k)!) + (2 + (-1)^k - (-1)^r) * (binomial(floor(r/2), floor(k/2)) + 2 * binomial(floor(r/2) - 1, floor(k/2) - 1)) * 3^(floor(r/2) - floor(k/2) - 1)). See Eq. (48) on p. 503 in the paper.
Letting k = 0 - 10, we get the eleven columns of Table 2 on p. 501 of Cyvin et al. (1996). (We need r >= max(k, 2) because the number of hexagons r should be greater than or equal to the number of converted polygons k.)
(End)

			Triangle begins (rows start at n = 2 and columns at k = 0):
     1,    1,     1;
     2,    3,     3,     1;
     4,    7,     9,     3,     1;
    10,   23,    29,    16,     5,    1;
    25,   69,    99,    62,    27,    5,    1;
    70,  229,   351,   275,   132,   39,    7,   1;
   196,  731,  1249,  1121,   643,  221,   55,   7,  1;
   574, 2385,  4437,  4584,  2997, 1278,  367,  72,  9, 1;
  1681, 7657, 15597, 18012, 13458, 6678, 2322, 540, 93, 9, 1;
  ...

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.

Column k = 0 is A001998. Column k = 3 is A323941.

For the element T(n, k) in row n >= 2 and column k >= 0 (such that max(k, 2) <= n), we have T(n, k) = I(r = n, k), where I(r, k) is given above in the comments. - Petros Hadjicostas, May 26 2019

Name edited by Petros Hadjicostas, May 26 2019

A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 97, 106, 114, 123, 133, 142, 152, 162, 173, 184, 195, 207, 219, 231, 244, 257, 270, 284, 297, 312, 326, 341, 357, 372, 388, 404, 421, 438, 455, 473, 491, 509, 528, 547, 566

Offset: 0

Jonathan Vos Post, Feb 27 2007

Kurz proved the polyomino equivalent of this conjecture as A122133 and abstracts: "In this article we prove a conjecture of Bezdek, Brass and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n."

			a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.

Colin Barker, Table of n, a(n) for n = 0..1000
Sascha Kurz, Convex hulls of polyominoes, arXiv:math/0702786 [math.CO], Feb 26 2007. See conjecture 2, p. 12.
Eric Weisstein's World of Mathematics, Polyhex.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000228, A036359, A002216, A005963, A001998, A018190, A001207, A057973, A122133.

Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* Harvey P. Dale, Apr 11 2012 *)

concat(0, Vec(x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)) + O(x^50))) \\ Colin Barker, Oct 13 2016

a(n) = (n^2 + 14*n/3 + 1)\6 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = floor((n^2 + 14*n/3 + 1)/6).

G.f.: x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)). - Colin Barker, Oct 13 2016

More terms from Harvey P. Dale, Apr 11 2012

Offset changed to 0 by Colin Barker, Oct 13 2016

A323944 Irregular triangle read by rows: Numbers of unbranched k-5-catafusenes.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 4, 8, 12, 5, 2, 10, 29, 48, 36, 15, 3, 25, 95, 193, 185, 114, 32, 6, 70, 329, 757, 933, 706, 316, 80, 10, 196, 1094, 2896, 4239, 3960, 2304, 866, 176, 20, 574, 3659, 10834, 18468, 20313, 14787, 7184, 2238, 408, 36, 1681, 12029, 39697, 76788, 97740, 84672, 51060, 20929, 5688, 896, 72

Offset: 2

N. J. A. Sloane, Feb 09 2019

			Triangle begins:
1, 1, 1,
2, 3, 3, 1,
4, 8, 12, 5, 2,
10, 29, 48, 36, 15, 3,
25, 95, 193, 185, 114, 32, 6,
70, 329, 757, 933, 706, 316, 80, 10,
196, 1094, 2896, 4239, 3960, 2304, 866, 176, 20,
574, 3659, 10834, 18468, 20313, 14787, 7184, 2238, 408, 36,
1681, 12029, 39697, 76788, 97740, 84672, 51060, 20929, 5688, 896, 72,
...

First column is A001998.

cp's OEIS Frontend

A056332 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of three different colors.

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A038766 Triangle giving number of unbranched catapolytetragons, read by rows.

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A323942 Irregular triangle read by rows giving the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes.

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A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

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PARI

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A323944 Irregular triangle read by rows: Numbers of unbranched k-5-catafusenes.

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