A002986 -id:A002986 - cp's OEIS frontend

A134818 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most triple edges.

1, 3, 9, 37, 146, 772, 4449, 30307, 228605, 1921464, 17652327, 176162548, 1893738334, 21806975279, 267636988052, 3486370839295, 48029272657002, 697542580286159, 10649954607360119, 170508064788069346, 2856122791685125616, 49951625299057923405

Offset: 1

David Consiglio, Jr., Jan 28 2008

From Natan Arie Consigli, May 29 2017: (Start)
Original name was "Number of hydrocarbon structures that can be drawn (excluding stereoisomers)" but this has been replaced with a mathematical definition which is more consistent with the terms of the sequence and the program.
In chemical terms this counts the following, given n carbon atoms:
- carbon allotropes;
- aliphatic hydrocarbons;
- resonance structures of graphically non-equivalent anti-aromatic and aromatic hydrocarbons.
Some molecules are theoretical and may or may not exist.
(End)
Computed over a period of several years and confirmed using the Molgen program.
Terms for n = 8,9,10 calculated using an exhaustive algorithm and Nauty. The algorithm correctly found the 7 known terms and the known acyclic hydrocarbons (up to n=10, see A002986) were extracted from the results correctly. - Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
Except for a(2), the same as A289157. The extra graph in A289157 is the 4-regular graph on 2 nodes. - Andrew Howroyd, Mar 20 2020

			For n = 2 there are a(2) = 3 structures that can be drawn with 2 carbons (ethane, ethene, and ethyne).
For n = 7 there are a(7) = 4449 structures that can be drawn with 7 carbons.

Brendan McKay, Nauty
Molgen, Publications

Cf. A134819 gives the number of possible structures, broken down by units of unsaturation.
Cf. A002986 (non-cyclic hydrocarbons).
Cf. A121941, A289157, A289158, A303033.

nauty
```
geng -c -D4 ${n} -q | multig -m3 -D4 -u
```

a(n) = A289157(n) for n > 2. - Andrew Howroyd, Mar 20 2020

a(8)-a(10) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
a(11) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
a(12) sent by David Consiglio, Jr., Apr 23 2008
a(12) corrected, a(13) and a(14) added - David Consiglio, Jr. Nov 03 2011
a(15)-a(17) computed using nauty by Sean A. Irvine, Jan 19 2015
New name from Natan Arie Consigli, May 29 2016
a(18)-a(22) from Andrew Howroyd, Mar 20 2020

A000631 Number of ethylene derivatives with n carbon atoms.

Original entry on oeis.org

1, 1, 3, 5, 13, 27, 66, 153, 377, 914, 2281, 5690, 14397, 36564, 93650, 240916, 623338, 1619346, 4224993, 11062046, 29062341, 76581151, 202365823, 536113477, 1423665699, 3788843391, 10103901486, 26995498151, 72253682560, 193706542776

Offset: 2

N. J. A. Sloane

nonn

Number of structural isomers of alkenes C_n H_{2n} with n carbon atoms.
Number of unicyclic graphs of n nodes where a double-edge replaces the cycle, [A217781], end-points of the double-edge of out-degrees <= 2, other nodes having out-degrees <= 3.
Number of rooted trees on n+1 nodes where the root has degree 2, the 2 children of the root have out-degrees <= 2, and the other nodes have out-degrees <= 3.
See illustration of initial terms. - Washington Bomfim, Nov 30 2020

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 2..500
Washington Bomfim, Illustration of initial terms
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1) (1934), 157-157.
C.-W. Lam, A Mathematical Relationship between the Number of Isomers of Alkenes and Alkynes: A Result Established from the Enumeration of Isomers of Alkenes from Alky Biradicals, J. Math. Chem., 23, 421 (1998).
R. J. Mathar, Illustration for graphs up to 6 carbons
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
R. C. Read, Some recent results in chemical enumeration, Preprint, circa 1972. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 28.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Index entries for sequences related to rooted trees

Cf. A000642, A000598, A027852 (out-degrees of nodes not limited).

\\ Here G(n) is A000598 as g.f., h is A000642.
seq(n)={my(g=G(n), h=(subst(g, x, x^2) + g^2)/2); Vec(subst(h, x, x^2) + h^2)/2} \\ Andrew Howroyd, Dec 01 2020

a(n) = b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b(n/2)(b(n/2) + 1)/2 when n is even or b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b((n-1)/2)b((n + 1)/2) when n is odd, where b(n) = A000642(n). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
a(n) = Sum_{k=1..(n-1)/2}( f(k) * f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2 + f(n/2) ) / 2 where f(n) = A000642(n+1). - Washington Bomfim, Nov 29 2020
G.f.: (g(x^2) + g(x)^2)/2 where x*g(x) is the g.f. of A000642. - Andrew Howroyd, Dec 01 2020

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

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)

A036672 Number of stereoisomers of acyclic hydrocarbons with n carbon atoms.

Original entry on oeis.org

1, 3, 4, 13, 31, 109, 372, 1446, 5714, 23791, 100827, 438019, 1931818, 8648820, 39178079, 179383748, 828905252, 3861958783, 18125392905, 85631735301, 406977645228, 1944737525915, 9338989516911, 45051405221284, 218236995129380, 1061256971559421

Offset: 1

N. J. A. Sloane

nonn

Comment from Sean A. Irvine, edited by Natan Arie Consigli, Dec 26 2016 : (Start)
This is the counting series for the hypothetical stereo-isomers of all acyclic hydrocarbons that satisfy the octet rule.
A036673 is the variant with triple bonds excluded.
A002986 doesn't count stereoisomers.
The reference gives a three-variable generating function and cycle-index over A4 which can produce both these sequences. There are also dependencies on earlier generating functions.
(End)
Read has incorrect a(10)=27100. - Sean A. Irvine, Nov 20 2020

			From _M. F. Hasler_, Dec 26 2016: (Start)
For n = 1, there is only a(1) = 1 possibility, CH4.
For n = 2, one has C2H6 (ethane, H3C-CH3), C2H4 (ethylene, H2C=CH2 with a double bond), C2H2 (ethyne, HC≡CH, triple bond), whence a(2) = 3.
For n = 3, one has C3H8 (H3C-CH2-CH3), C3H6 (H2C=CH-CH3, propene), and two C3H4 (H2C=C=CH2, propadiene, and HC≡C-CH3: methylacetylene), thus a(3) = 4. Cyclic molecules like cyclopropane C3H6 and cyclopropropene C3H4 are excluded. (End)
From _Natan Arie Consigli_, Dec 26 2016: (Start)
For n = 4, we have butane, isobutane, 1-butene, cis/trans-2-butene, buta-1,2-diene, buta-1,3-diene, butatriene, isobutylene, but-1-yne, but-2-yne, diacetylene, but-1-en-3-yne.
For n = 5 we have:
- 3 alkanes: pentane, methylbutane and neopentane.
- 17 alkenes: 1-pentene, (E/Z)-2-pentene, 1,2-pentadiene, (E/Z)-1,3-pentadiene, 1,4-pentadiene, 1,2,3-petatriene, penta-1,2,4-triene, pentatetraene, 2-methylbut-1-ene, 2-methylbut-2-ene, 3-methylbut-1-ene, isoprene, 3-methylbuta-1,2-diene, (R/S)-penta-2,3-diene.
-11 alkynes: 1-pentyne, 2-pentyne, pent-1-en-4-yne, (E/Z)-pent-3-en-1-yne, penta-1,2-dien-4-yne, penta-1,4-diyne, penta-1,3-diyne, pent-1-en-3-yne, 3-methylbut-1-yne, 2-methylbut-1-en-3-yne. (End)

Sean A. Irvine, Correction of A036672(10) and A036673(10), 2020
Sean A. Irvine, Java program (github)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 60.
Wikipedia, Stereoisomerism

Cf. A000598-A000628, A002986, A010372, ..., A086194, ... A134818, ..., A173289, ..., A261336.

Cf. A107984, A036673.

a(10) corrected and more terms from Sean A. Irvine, Nov 20 2020

cp's OEIS Frontend

A134818 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most triple edges.

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nauty

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A000631 Number of ethylene derivatives with n carbon atoms.

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

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A036672 Number of stereoisomers of acyclic hydrocarbons with n carbon atoms.

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A117850 Number of hydrocarbons with n carbon atoms.

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