Displaying 1-10 of 12 results found.
Number of mirror-symmetrical edge-rooted tree-like octagonal systems.
+10
8
1, 1, 3, 4, 15, 23, 94, 155, 661, 1139, 4983, 8844, 39362, 71360, 321561, 592361, 2694421, 5025849, 23029195, 43388208, 199990961, 379900479, 1759636142, 3365582261, 15652514944, 30112397278, 140531706444, 271707661708
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134(1) (1997), 55-70. [The index of summation in Eq. (15), p. 60, should start at i = 0, not at i = 1. - Petros Hadjicostas, Jul 30 2019]
FORMULA
G.f. V=V(x) satisfies x(x-2)V^3 + 2(x^2-3x+1)V^2 + (-x^2-3x+2)V - x(x+2) = 0.
Let U(0) = 1 and U(n) = A036758(n) for n >= 1. Let also a(0) = a(1) = 1 (even though the offset for the current sequence is 1 as it is done in Table II (p. 61) in Cyvin et al. (1997) and in Eq. (5), p. 195, in Brunvoll et al. (1997)). Then
a(n) = Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n even >= 2, and
a(n) = U((n-1)/2) + Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n odd >= 3.
This is Eq. (15), p. 60, in Cyvin et al. (1997), but we have corrected the lower index of summation (from i = 1 to i = 0).
(End)
MAPLE
F := (2+3*V+6*V^2+2*V^3-(V+2)*sqrt(1+4*V+8*V^2+4*V^4))/2/(V^3+2*V^2-V-1): Order := 40: S := solve(series(F, V)=x, V);
PROG
(PARI) a(n)=if(n<1, 0, polcoeff(serreverse((2*x^3+6*x^2+3*x+2-(x+2)*sqrt(4*x^4+8*x^2+4*x+1+x*O(x^n)))/2/(x^3+2*x^2-x-1)), n)) /* Michael Somos, Mar 10 2004 */
Number of tree-like octagonal systems.
+10
5
1, 1, 3, 11, 56, 341, 2351, 17329, 133126, 1052813, 8511339, 70050568, 585226579, 4952310082, 42376979857, 366185395267, 3191787761573, 28036397185485, 247982793594183, 2207183342498485, 19756959439608022, 177765564883970415
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70.
FORMULA
G.f.: (2x + (2-7x)U(x) - (1+7x)U^2(x) + 2(2-x)V(x) + 4(1+x)U(x^2) + 3xU^2(x^2) + 2xU(x^4))/8, where U(x) is the g.f. of A036758 and V(x) is the g.f. of A036759. - Emeric Deutsch, May 04 2004
Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
+10
5
0, 5, 25, 155, 1080, 8085, 63525, 516790, 4315805, 36786385, 318736105, 2799049985, 24857641900, 222861398060, 2014418084860, 18337277269475, 167961106916065, 1546879330598945, 14315792338559005, 133065134882334095, 1241694764334690820, 11628016504072124555, 109243880617142972435
COMMENTS
Quoting from p. 59 in Cyvin et al. (1997): "When an octagon is rooted at an edge ... then either (a) one branch can be attached in five directions at a time, (b) two branches can be attached in six ways, or (c) three branches in one way. Let the numbers of these kinds of systems be denoted by (a) U_r^*, (b) U_r^{**}, and (c) U_r^{***}, respectively."
Here r is "the number of octagons or eight-membered rings" in an edge-rooted catapolygon (here, catapolyoctagon). A catapolyoctagon is a "catacondensed polygonal system consisting of octagons" (where "catacondensed" means it has no internal vertices).
On p. 59 in Cyvin et al. (1997), the total number of edge-rooted catapolyoctagons (each with r octagons) is denoted by U_r, and we have U_r = U_r^* + U_r^{**} + U_r^{***} for r >= 2.
We have U_r = A036758(r), U_r^* = a(r) (current sequence), U_r^{**} = A121113(r), and U_r^{***} = A121114(r) for r >= 1. For the current sequence, we have a(r) = U_r^* = 5*U_{r-1} = 5* A036758(r-1) for r >= 2 with a(1) = U_1^* = 0. The ultimate purpose of these calculations (in the paper by Cyvin et al. (1997)) is the calculation of I_r = A036760(r), which is the "number of nonisomorphic free (unrooted) catapolyoctagons when r is given." These catapolyoctagons "represent a class of polycyclic conjugated hydrocarbons, C_{6r+2} H_{4r+4}" (see p. 57 in Cyvin et al. (1997)). (End)
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70; see pp. 59-61.
MAPLE
Order := 30: S := solve(series(G/(1+5*G+6*G^2+G^3), G)=x, G);
Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
+10
5
0, 0, 6, 60, 522, 4452, 38130, 329832, 2884056, 25476936, 227145654, 2041930920, 18490834362, 168537705300, 1545096559812, 14238592913328, 131826509242650, 1225645805016864, 11438847800351118, 107128560124123524, 1006475582377759578, 9483340466106708180, 89593844489912910294
COMMENTS
Quoting from p. 59 in Cyvin et al. (1997): "When an octagon is rooted at an edge ... then either (a) one branch can be attached in five directions at a time, (b) two branches can be attached in six ways, or (c) three branches in one way. Let the numbers of these kinds of systems be denoted by (a) U_r^*, (b) U_r^{**}, and (c) U_r^{***}, respectively."
Here r is "the number of octagons or eight-membered rings" in an edge-rooted catapolygon (here, catapolyoctagon). A catapolyoctagon is a "catacondensed polygonal system consisting of octagons" (where "catacondensed" means it has no internal vertices).
On p. 59 in Cyvin et al. (1997), the total number of edge-rooted catapolyoctagons (each with r octagons) is denoted by U_r, and we have U_r = U_r^* + U_r^{**} + U_r^{***} for r >= 2.
We have U_r = A036758(r), U_r^* = A121112(r), U_r^{**} = a(r) (current sequence), and U_r^{***} = A121114(r) for r >= 1. For the current sequence, we have a(r) = U_r^{**} = 6*Sum_{i = 1.. r-2} U(i) * U(r-1-i) for r >= 3, where U(r) = A036758(r), with a(1) = a(2) = 0. See Eq. (12) in Cyvin et al. (1997). The ultimate purpose of these calculations (in the paper by Cyvin et al. (1997)) is the calculation of I_r = A036760(r), which is the "number of nonisomorphic free (unrooted) catapolyoctagons when r is given." These catapolyoctagons "represent a class of polycyclic conjugated hydrocarbons, C_{6r+2} H_{4r+4}" (see p. 57 in Cyvin et al. (1997)). The g.f.'s of the sequences U, U^*, U^{**}, and U^{***} appear also in Eqs. (2) and (3) on p. 194 in Brunvoll et al. (1997).
(End)
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70; see pp. 59-61.
FORMULA
a(r) = 6*Sum_{i = 1.. r-2} U(i) * U(r-1-i) for r >= 3, where U(r) = A036758(r), with a(1) = a(2) = 0. - Petros Hadjicostas, Jul 30 2019
MAPLE
Order := 30; S := solve(series(G/(G^3 + 6*G^2 + 5*G + 1), G) = x, G);
Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
+10
5
0, 0, 0, 1, 15, 168, 1703, 16539, 157416, 1483900, 13928238, 130547475, 1223803350, 11484513612, 107940809223, 1016351200410, 9588249961074, 90633332095992, 858386837556696, 8145257860480545, 77432954101974513, 737419153249761752, 7034562802431438771, 67214038308803342715
COMMENTS
Quoting from p. 59 in Cyvin et al. (1997): "When an octagon is rooted at an edge ... then either (a) one branch can be attached in five directions at a time, (b) two branches can be attached in six ways, or (c) three branches in one way. Let the numbers of these kinds of systems be denoted by (a) U_r^*, (b) U_r^{**}, and (c) U_r^{***}, respectively."
Here r is "the number of octagons or eight-membered rings" in an edge-rooted catapolygon (here, catapolyoctagon). A catapolyoctagon is a "catacondensed polygonal system consisting of octagons" (where "catacondensed" means it has no internal vertices).
On p. 59 in Cyvin et al. (1997), the total number of edge-rooted catapolyoctagons (each with r octagons) is denoted by U_r, and we have U_r = U_r^* + U_r^{**} + U_r^{***} for r >= 2.
We have U_r = A036758(r), U_r^* = A121112(r), U_r^{**} = A121113(r), and U_r^{***} = a(r) (current sequence) for r >= 1. For the current sequence, we have a(r) = U_r^{***} = Sum_{i = 1..r-3} U(i) * Sum_{j = 1..r-i-2} U(j) * U(r-1-i-j) for r >= 4, where U(r) = A036758(r), with a(1) = a(2) = a(3) = 0. See Eq. (13) on p. 59 in Cyvin et al. (1997). The ultimate purpose of these calculations (in the paper by Cyvin et al. (1997)) is the calculation of I_r = A036760(r), which is the "number of nonisomorphic free (unrooted) catapolyoctagons when r is given." These catapolyoctagons "represent a class of polycyclic conjugated hydrocarbons, C_{6r+2} H_{4r+4}" (see p. 57 in Cyvin et al. (1997)). The g.f.'s of the sequences U, U^*, U^{**}, and U^{***} appear also in Eqs. (2) and (3) on p. 194 in Brunvoll et al. (1997).
(End)
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70; see pp. 59-61.
FORMULA
a(r) = Sum_{i = 1..r-3} U(i) * Sum_{j = 1..r-i-2} U(j) * U(r-1-i-j) for r >= 4, where U(r) = A036758(r), with a(1) = a(2) = a(3) = 0. - Petros Hadjicostas, Jul 30 2019
MAPLE
Order := 30;
S := solve(series(G/(G^3 + 6*G^2 + 5*G + 1), G) = x, G);
Related to enumeration of rooted catapolyoctagons (see Cyvin reference for precise definition).
+10
3
0, 1, 10, 87, 742, 6355, 54972, 480676, 4246156, 37857609, 340321820, 3081805727, 28089617550, 257516093302, 2373098818888, 21971084873775, 204274300836144, 1906474633391853, 17854760020687254, 167745930396293263, 1580556744351118030, 14932307414985485049, 141420575129236412394
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, Table III, W_r.
Related to enumeration of rooted catapolyoctagons (see Cyvin reference for precise definition).
+10
1
0, 0, 1, 15, 168, 1703, 16539, 157416, 1483900, 13928238, 130547475, 1223803350, 11484513612, 107940809223, 1016351200410, 9588249961074, 90633332095992, 858386837556696, 8145257860480545, 77432954101974513, 737419153249761752, 7034562802431438771, 67214038308803342715
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, Table III, X_r.
Related to enumeration of rooted catapolyoctagons (see Cyvin reference for precise definition).
+10
0
0, 0, 0, 1, 20, 274, 3224, 35119, 366152, 3717126, 37101780, 366240435, 3588644268, 34989880068, 340028122980, 3297155317612, 31927270515392, 308907360676012, 2987556153506904, 28890414890866473, 279405472916899476, 2702885116291760036, 26156597304381215188
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, Table III, Y_r.
Related to enumeration of rooted catapolyoctagons (see Cyvin reference for precise definition).
+10
0
0, 0, 0, 1, 2, 12, 24, 113, 232, 1008, 2122, 8891, 19110, 78540, 171694, 697700, 1546552, 6240064, 13992408, 56192581, 127231246, 509314256, 1162747842, 4644046225, 10677793584
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70.
Related to enumeration of free catapolyoctagons (see Cyvin reference for precise definition).
+10
0
1, 1, 6, 32, 245, 1926, 16119, 137475, 1194854, 10517058, 93593287, 840500847, 7607643154, 69331324533, 635651786601, 5858956542866, 54260363841729, 504655054951887, 4711672806323948, 44143665936691049, 414896145593829741, 3910842418588639201
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, Tbale IV, P_r.
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