A129012 -id:A129012 - cp's OEIS frontend

A129013 A129012(3n).

Original entry on oeis.org

1, 2, 7, 27, 86, 264, 749, 2022

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Cf. A129012-A129021.

A129014 A129012(3n+1).

Original entry on oeis.org

1, 3, 12, 38, 128, 373, 1055, 2765

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Cf. A129012-A129021.

A129016 A129012 with the third term from each triple omitted.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 27, 38, 86, 128, 264, 373, 749, 1055, 2022, 2765

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Cf. A129012-A129021.

A129021 A129017 with the third term from each triple omitted.

Original entry on oeis.org

1, 1, 2, 4, 12, 19, 46, 70, 162, 239, 504, 726, 1471, 2062

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Jerry Ray Dias, Constant-isomer benzenoid series and their polyradical subsets, Theoretica Chimica Acta 81(3) (1991), 125-138; see Table 2 on p. 129.
Jerry Ray Dias, Notes on constant-isomer series, J. Chem. Inf. Comput. Sci. 33 (1993), pp. 117-127; see p. 123.
Jerry Ray Dias, Graph theoretical invariants and elementary subgraphs of polyhex and polypent/polyhex systems of chemical relevance, Discr. App. Math. 67 (1-3) (1996), pp. 79-114; see p. 98.

Cf. A129012-A129021.

Name edited by Petros Hadjicostas, Nov 17 2019

A306334 a(n) is the number of different linear hydrocarbon molecules with n carbon atoms.

Original entry on oeis.org

1, 3, 4, 10, 18, 42, 84, 192, 409, 926, 2030, 4577, 10171, 22889, 51176, 115070, 257987, 579868, 1301664, 2925209, 6569992, 14763529, 33166848, 74527233, 167446566, 376253517, 845401158, 1899609267, 4268309531, 9590827171, 21550227328, 48422972296, 108805058758

Offset: 1

Vincent Champain, Feb 08 2019

Linear hydrocarbons are molecules made of carbon (C) and hydrogen (H) atoms organized without cycles.
a(n) <= A002986(n) because molecules can be acyclic but not linear (i.e., including carbon atoms bonded with more than two other carbons).
From Petros Hadjicostas, Nov 16 2019: (Start)
We prove Vaclav Kotesovec's conjectures from the Formula section. Let M = [[0,0,1], [0,1,1], [1,1,1]], X(n) = M^(n-2), and Y(n) = M^(floor(n/2)-2) = X(floor(n/2)) (with negative powers indicating matrix inverses). Let also, t_1 = [1,1,1]^T, t_2 = [1,2,2]^T, and t_3 = [1,2,3]^T. In addition, define b(n) = (1/2)*(t_1^T X(n) t_1) and c(n) = (1/2)*(t_3^T Y(n) t_1) if n is even and = (1/2)*(t_2^T Y(n) t_1) if n is odd.
We have a(n) = b(n) + c(n) for n >= 1. Since the characteristic polynomial of Vaclav Kotesovec's recurrence is x^9 - 2*x^8 - 3*x^7 + 5*x^6 + x^5 + 2*x^3 - 3*x^2 - x + 1 = g(x)*g(x^2), where g(x) = x^3 - 2*x^2 - x + 1, to prove his first conjecture, it suffices to show that b(n) - 2*b(n-1) - b(n-2) + b(n-3) = 0 (whose characteristic polynomial is g(x)) and c(n) - 2*c(n-2) - c(n-4) + c(n-6) = 0 (whose characteristic polynomial is g(x^2)).
Note that 2*b(n) = A006356(n-1) for n >= 1. (See the comments by L. Edson Jeffery and R. J. Mathar in the documentation of that sequence.) Also, 2*c(2*n) = A006356(n) and 2*c(2*n-1) = A006054(n+1) for n >= 1.
Properties of the polynomial g(x) = x^3 - 2*x^2 - x + 1 and its roots were studied by Witula et al. (2006) (see Corollary 2.4). This means that a(n) can essentially be expressed in terms of exp(I*2*Pi/7), but we omit the discussion. See also the comments for sequence A006054.
The characteristic polynomial of matrix M is g(x). By the Cayley-Hamilton theorem, 0 = g(M) = M^3 - 2*M^2 - M + I_3, and thus, for n >= 5, X(n) - 2*X(n-1) - X(n-2) + X(n-3) = M^(n-2) - 2*M^(n-3) - M^(n-4) + M^(n-5) = 0. Pre-multiplying by (1/2)*t_1^T and post-multiplying by t_1, we get that b(n) - 2*b(n-1) - b(n-2) + b(n-3) = 0 for n >= 5.
Similarly, for n >= 10, Y(n) - 2*Y(n-2) - Y(n-4) + Y(n-6) = X(floor(n/2)) - 2*X(floor((n-2)/2)) - X(floor((n-4)/2)) + X(floor((n-6)/2)) = X(floor(n/2)) - 2*X(floor(n/2) - 1) - X(floor(n/2) - 2) + X(floor(n/2) - 3) = 0. Pre-multiplying by (1/2)*t_3^T (when n is even) or by (1/2)*t_2^T (when n is odd), and post-multiplying by t_1, we get c(n) - 2*c(n-2) - c(n-4) + c(n-6) = 0 for n >= 10.
Since the characteristic polynomial of Vaclav Kotesovec's recurrence is g(x)*g(x^2), which is a polynomial of degree 9, the denominator of the g.f. of the sequence (a(n): n >= 1) should be x^9*g(1/x)*g(1/x^2) = (1 - 2*x - x^2 + x^3)*(1 - 2*x^2 - x^4 + x^6), as Vaclav Kotesovec conjectured below. The numerator of Vaclav Kotesovec's g.f. can be easily derived using the initial conditions (from a(1) = 1 to a(9) = 409). (End)

			For n=1, there is one possibility: CH4.
For n=2, there are 3 solutions: CHCH, CH3CH3, CH2CH2.
For n=3, there are 4 solutions: CHCCH3, CH2CCH2, CH3CHCH2, CH3CH2CH3.
For n=6, there are 42 solutions: CH3CH2CHCHCCH, CH3CH2CHCHCH2CH3, CH2CHCCCHCH2, CH2CHCHCHCH2CH3, CH2CHCHCHCCH, CH2CCCCHCH3, CHCCCCHCH2, CH3CHCHCHCHCH3, CHCCHCHCCH, CH2CCCCCH2, CH3CCCH2CH2CH3, CH3CCCCCH3, CH3CH2CH2CH2CH2CH3, CH2CHCHCHCHCH2, CH2CCHCH2CHCH2, CH3CHCCCHCH3, CHCCH2CH2CH2CH3, CHCCH2CH2CCH, CH3CCCH2CHCH2, CH2CCCHCH2CH3, CH2CCCHCCH, CHCCH2CCCH3, CHCCH2CHCCH2, CH3CH2CH2CH2CHCH2, CH2CHCHCCHCH3, CH3CH2CCCH2CH3, CH2CHCH2CH2CHCH2, CH2CHCHCCCH2, CH3CHCCHCH2CH3, CH3CH2CH2CHCHCH3, CH3CHCCHCCH, CHCCH2CH2CHCH2, CH3CHCHCCCH3, CH2CCHCCCH3, CH3CHCHCHCCH2, CHCCCCH2CH3, CH2CHCH2CHCHCH3, CH2CCHCHCCH2, CHCCCCCH, CH2CCHCH2CH2CH3, CH3CH2CCCHCH2, CHCCH2CHCHCH3.

Vincent Champain, Table of n, a(n) for n = 1..1000
L. Edson Jeffery, Danzer matrices (unit-primitive matrices). [It contains a discussion of a generalization of the matrix M that appears in the formula for a(n). See basis D_7.]
Wikipedia, Cayley-Hamilton theorem.
R. Witula, D. Slota, and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq. 9 (2006), Article 06.4.3. [See Corollary 2.4 and the discussion about the polynomial p_7(x) and its roots. This essentially proves that a(n) can be expressed in terms of exp(I*2*Pi/7).]
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-1,0,-2,3,1,-1).

Other hydrocarbon related sequences: A002986, A018190, A129012.

Cf. A006054, A006356.

with(LinearAlgebra):
M := Matrix([[0, 0, 1], [0, 1, 1], [1, 1, 1]]):
X := proc(n) MatrixPower(M, n - 2): end proc:
Y := proc(n) MatrixPower(M, floor(1/2*n) - 2): end proc:
a := proc(n) `if`(n < 4, [1,3,4][n], 1/2*(add(add(X(n)[i, j], i = 1..3), j = 1..3) + add(add(Y(n)[i, j]*min(j, 3 - (n mod 2)), i = 1..3), j = 1..3))):
     end proc:
seq(a(n), n=1..40); # Petros Hadjicostas, Nov 17 2019

M = {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}};
X[n_] := MatrixPower[M, n - 2];
Y[n_] := MatrixPower[M, Floor[1/2*n] - 2];
a[n_] := If[n < 4, {1, 3, 4}[[n]], 1/2*(Sum[Sum[X[n][[i, j]], {i, 1, 3}], {j, 1, 3}] + Sum[Sum[Y[n][[i, j]]*Min[j, 3 - Mod[n, 2]], {i, 1, 3}], {j, 1, 3}])];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 16 2023, after Petros Hadjicostas *)

from numpy import array as npa
from numpy.linalg import matrix_power as npow
def F(n):
     if n<4: return([0,1,3,4][n])
     m=npa([[0,0,1],[0,1,1],[1,1,1]],dtype=object)
     m2=npow(m,n//2-2)
     return((sum(sum(npow(m,n-2)))+sum(sum(m2[j]*min(j+1,3-(n&1)) for j in range(3))))//2)

a(n) = (1/2) * (Sum_{i,j = 1..3} X_{ij} + Sum_{i,j = 1..3} Y_{ij} * min(j, 3 - (n&1))), where M = [[0,0,1], [0,1,1], [1,1,1]], X = [X_{ij}: i,j = 1..3] = M^(n-2), and Y = [Y_{ij}: i,j = 1..3] = M^(floor(n/2)-2)) for n >= 1 (with negative powers indicating matrix inverses). [Edited by Petros Hadjicostas, Nov 16 2019]
Conjectures from Vaclav Kotesovec, Feb 12 2019: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 5*a(n-3) - a(n-4) - 2*a(n-6) + 3*a(n-7) + a(n-8) - a(n-9), for n >= 10.
G.f.: (1 - x - 2*x^2 - x^4 + 2*x^5 + x^6 - x^7) / ((1 - 2*x - x^2 + x^3)*(1 - 2*x^2 - x^4 + x^6)) - 1. (End) [These conjectures are true. See my comments above. - Petros Hadjicostas, Nov 17 2019]
From Petros Hadjicostas, Nov 17 2019: (Start)
a(2*n) = (1/2)*(A006356(2*n-1) + A006356(n)).
a(2*n-1) = (1/2)*(A006356(2*n-2) + A006054(n+1)). (End)

A129017 Isomer numbers for the constant-isomer series: the monradical, diradical, triradical, tetraradical, etc. series.

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 12, 19, 19, 46, 70, 70, 162, 239, 239, 504, 726, 726, 1471, 2062, 2062

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

The terms occur in groups of three, X, X, Y.
From Petros Hadjicostas, Nov 17 2019: (Start)
When we count diradical isomers, the starting compounds for each constant-isomer series are as follows (with the number of diradical isomers inside parentheses): C22H12 (1), C30H14 (1), C40H16 (1), C50H18 (2), C62H20 (4), C76H22 (4), C90H24 (12), C106H26 (19), C124H28 (19), C142H30 (46), C162H32 (70), C184H34 (70), C206H36 (162), C230H38 (239), C256H40 (239), C282H42 (504), C310H44 (726), C340H46 (726), C370H48 (1471), C402H50 (2062), C436H52 (2062).
For some mysterious reason, the ground compounds of each series, which are listed above, obey the general formula C_{2*b(s)} H_{2*s}, where b(s) = A096777(s), for s = 6, 7, ..., 26.
Given a ground compound in a constant-isomer series of compounds, the series is determined by the operator P(C_n H_s) -> C_{n + 2*s + 6} H_{s + 6}. For example, the series corresponding to the C22H12 is C22H12 -> C52H18 -> C94H24 -> C148H30 -> C214H36 -> ...
As it can be seen in Dias (1996), the same numbers appear for the number of monoradical isomers for odd-carbon compounds starting with C13H9. See also Table 1 in Dias (1991, p. 128). Here we have the following starting compounds for each constant-isomer series (with the number of monoradical isomers in parentheses): C13H9 (1), C19H11 (1), C27H13(1), C35H15 (2), C45H17 (4), C57H19 (4), C69H21 (12), C83H23 (19), C99H25 (19), C115H27 (46), ...
For additional interpretations of these numbers (e.g., in terms of tetraradicals), see the equations and theory in Dias (1993).
The "base formulas for the smallest one-isomer polyradicals" appear in Section 7.6 in Dias (1996), which explains why we begin with C22H12 for the number of diradical isomers and with C13H9 for the number of monoradicals.
(End)

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Jerry Ray Dias, Constant-isomer benzenoid series and their polyradical subsets, Theoretica Chimica Acta 81(3) (1991), 125-138; see Table 2 on p. 129.
Jerry Ray Dias, Notes on constant-isomer series, J. Chem. Inf. Comput. Sci. 33 (1993), pp. 117-127; see p. 123.
Jerry Ray Dias, Graph theoretical invariants and elementary subgraphs of polyhex and polypent/polyhex systems of chemical relevance, Discr. App. Math. 67 (1-3) (1996), pp. 79-114; see p. 98 (same numbers for the number of monoradical isomers for odd-carbon compounds starting with C13H9).

Cf. A096777, A129012-A129021.

A129018 A129017(3n).

Original entry on oeis.org

1, 2, 12, 46, 162, 504, 1471

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Cf. A129012-A129021.

A129019 A129017(3n+1).

Original entry on oeis.org

1, 4, 19, 70, 239, 726, 2062

Offset: 0

N. J. A. Sloane, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007

nonn

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

Cf. A129012-A129021.

A129022 Isomer numbers for the fluorenoid/fluoranthenoid constant-isomer series.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 7, 7, 9, 17, 9, 31, 31, 41, 72, 41, 120, 120

Offset: 0

N. J. A. Sloane, May 10 2007

This allows a single pentagonal ring among otherwise hexagonal rings.
The terms occur in groups of 5: X, X, Y, Z, Y, ..., except for the 0th term (corresponding to the C5H5 series). [Edited by Petros Hadjicostas, Nov 17 2019]
From Petros Hadjicostas, Nov 17 2019: (Start)
From all the papers listed in the Links, the author of this sequence considers Dias (1991) as the basic one for this sequence.
The starting compounds for each constant-isomer series are as follows (with the number of isomers inside parentheses): C5H5 (1), C9H7 (1), C12H8 (1), C15H9 (2), C18H10 (3), C23H11 (2), C26H12 (7), C31H13 (7), C36H14 (9), C41H15 (17), C48H16 (9), C53H17 (31), C60H18 (31), C67H19 (41), C74H20 (72), C83H21 (41), C90H22 (120), C99H23 (120), C108H24 (...), C117H25 (...), C128H26 (...), where "..." means that the corresponding number of isomers is not listed in the papers (probably because they are not known yet).
Starting with each one of the above compounds, the corresponding constant-isomer series is generated by the operator P(C_n H_s) -> C_{n + 2*s + 5} H_{s + 5} successively. For example, the first series is C5H5 -> C20H10 -> C45H15 -> C80H20 -> ...
The starting compound of each constant-isomer series is C_n H_s, where n = s + 2 + 2*floor((1/10) * (s^2 - 7*s + 6)), for s = 5, 7, 8, 9, 10, 11, 12, ... (i.e., we skip s = 6). Thus, a(0) = 1 corresponds to s = 5 (i.e., C5H5), and for m >= 1, a(m) corresponds to s = m + 6, i.e., to C_n H_s, where n = s + 2 + 2*floor((1/10) * (s^2 - 7*s + 6)). See Cyvin et al. (1993, p. 233). (End)

J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D.H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396. [See Section 5.2.]

J. Brunvoll, B. N. Cyvin, S. J. Cyvin, G. Brinkmann, and J. Bornhoft, Enumeration of chemical isomers of polyclic conjugated hydrocarbons with six-membered and five-membered rings, Z. Naturforsch. 51a (1996), pp. 1073-1078.
S. J. Cyvin, B. N. Cyvin and J. Brunvoll, Graph-theoretical studies on fluoranthenoids and fluorenoids, J. Molec. Struct. (Theochem) 281(2-3) (1993), pp. 229-236.
Jerry Ray Dias, Series of fluorenoiod/fluroranthenoid hydrocarbons having a constant number of isomers, Chem. Phys. Lett. 185 (1-2) (1991), pp. 10-15. [See Table 2, p. 13.] (*)
Jerry Ray Dias, Deciphering the information content of chemical formulas: Chemical and structural characteristics and enumeration of indacenes, J. Chem. Inf. Comput. Sci. 32 (1992), pp. 203-209. [Table III, p. 206, gives the number of isomers for indacenoid hydrocarbon constant-isomer series that appear in A129023. The paper, however, contains a useful explanation of the whole theory.]
Jerry Ray Dias, Notes on constant-isomer series, J. Chem. Inf. Comput. Sci. 33 (1993), pp. 117-127.
Jerry Ray Dias, Graph theoretical invariants and elementary subgraphs of polyhex and polypent/polyhex systems of chemical relevance, Discr. App. Math. 67 (1-3) (1996), pp. 79-114. [This paper contains a clear mathematical definition of all the symbols used in the chemical papers. It is also contains a description of the so-called "aufbau principle".]
Jerry Ray Dias, Structure/formula informatics of isomeric sets of fluoranthemoid/fluorenoid and indacenoid hydrocarbons, Journal of Mathematical Chemistry 48(2) (2010), pp. 313-329. [It contains more recent information about this sequence and other similar sequences, A129012-A129021 and A129023.]

Cf. A129012-A129021, A129023.

A129023 Isomer numbers for the indacenoid constant-isomer series.

Original entry on oeis.org

1, 1, 1, 3, 6, 7, 6, 12, 28, 26, 28, 51

Offset: 0

N. J. A. Sloane, May 10 2007

nonn

This allows two single pentagonal rings among otherwise hexagonal rings.

The terms occur in groups of 4: X, Y, X, Y, ...

J. Brunvoll, B. N. Cyvin, S. J. Cyvin, G. Brinkmann and J. Bornhoft, Z. Naturforsch., 51a, 1073 (1996).
S. J. Cyvin, B. N. Cyvin and J. Brunvoll, J. Molec. Struct. (Theochem), 281, 229 (1993).
J. R. Dias, Chem. Phys. Lett., 185, 10 (1991).
J. R. Dias, J. Chem. Inf. Comput. Sci., 32, 203-209 (1992). (*)
J. R. Dias, J. Chem. Inf. Comput. Sci. 33, 117 (1993).
J. R. Dias, Discr. App. Math., 67, 79 (1996).
J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D.H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396. [See Section 5.2.]

Cf. A129012-A129022.

cp's OEIS Frontend

A129013 A129012(3n).

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A129014 A129012(3n+1).

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A129016 A129012 with the third term from each triple omitted.

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A129021 A129017 with the third term from each triple omitted.

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A306334 a(n) is the number of different linear hydrocarbon molecules with n carbon atoms.

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A129017 Isomer numbers for the constant-isomer series: the monradical, diradical, triradical, tetraradical, etc. series.

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A129018 A129017(3n).

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A129019 A129017(3n+1).

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A129022 Isomer numbers for the fluorenoid/fluoranthenoid constant-isomer series.

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A129023 Isomer numbers for the indacenoid constant-isomer series.

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