A000602 -id:A000602 - cp's OEIS frontend

A000200 Number of bicentered hydrocarbons with n atoms.

0, 0, 1, 0, 1, 1, 3, 3, 9, 15, 38, 73, 174, 380, 915, 2124, 5134, 12281, 30010, 73401, 181835, 452165, 1133252, 2851710, 7215262, 18326528, 46750268, 119687146, 307528889, 792716193, 2049703887, 5314775856, 13817638615, 36012395538

Offset: 0

N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)

Busacker and Saaty, Finite Graphs and Networks, 1965, p. 201 (they reproduce Cayley's mistakes).
A. Cayley, "On the mathematical theory of isomers", Phil. Mag. vol. 67 (1874), 444-447.
A. Cayley, "Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden...", Chem. Ber. 8 (1875), 1056-1059.
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..60
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678
Henry Bottomley, Illustration of initial terms of A000022, A000200, A000602
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
Index entries for sequences related to trees

A000200 = A000602 - A000022 for n>0.

Cf. A010373.

N := 45: for i from 1 to N do tt := t[ i ]-t[ i-1 ]; b[ i ] := series((tt^2+subs(z=z^2,tt))/2+O(z^(N+1)),z,200): od: i := 'i': bicent := series(sum(b[ i ],i=1..N),z,200); G000200 := bicent; A000200 := n->coeff(G000200,z,n);
# Maple code continues from A000022: bicentered == unordered pair of ternary trees of the same height:

n = 40; (* algorithm from Rains and Sloane *)
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S3[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)

A112410 Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 1, 5, 17, 56, 182, 573, 1792, 5533, 16977, 51652, 156291, 470069, 1407264, 4193977, 12451760, 36838994, 108656009, 319583578, 937634011, 2744720126, 8018165821, 23379886511, 68056985580, 197800670948, 574068309840, 1663907364480, 4816910618093, 13929036720057

Offset: 1

Jonathan Vos Post, Dec 08 2005

nonn

Such graphs are also referred to (e.g., by Hendrickson & Parks) as carbon skeletons with two rings, or bicyclic skeletons, although actual number of simple cycles in such graphs can exceed 2 (e.g., in the example). - Andrey Zabolotskiy, Nov 24 2017

Terms computed with nauty agree at least to a(20) with those computed by formula and sequences A125669, A125670, A125671, A305132. - Andrew Howroyd, May 26 2018

			The only such graph for n = 4 is:
o-o
|/|
o-o

Andrew Howroyd, Table of n, a(n) for n = 1..200
J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 2 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, this sequence, A112619, A112408, A112424, A112425, A112426, A112442.
Cf. A121941 (any number of edges), A006820 (2n edges).
Cf. A125669, A125670, A125671, A305132.

for n in {4..15}; do geng -c -D4 ${n} $((n+1)):$((n+1)) -u; done # Andrey Zabolotskiy, Nov 24 2017

a(n) = A125669(n) + A125670(n) + A125671(n) + A305132(n). - Andrew Howroyd, May 26 2018

Corrected offset and new name from Andrey Zabolotskiy, Nov 20 2017
a(20) corrected by Andrey Zabolotskiy and Andrew Howroyd, May 26 2018
Terms a(21) and beyond from Andrew Howroyd, May 26 2018

A180148 a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.

Original entry on oeis.org

2, 5, 17, 56, 185, 611, 2018, 6665, 22013, 72704, 240125, 793079, 2619362, 8651165, 28572857, 94369736, 311682065, 1029415931, 3399929858, 11229205505, 37087546373, 122491844624, 404563080245, 1336181085359, 4413106336322, 14575500094325, 48139606619297

Offset: 0

Johannes W. Meijer, Aug 13 2010

Inverse binomial transform of A052961 (without the leading 1).

For n >= 1, also the number of matchings in the n-alkane graph. - Eric W. Weisstein, Jul 14 2021

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Alkane Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
Index entries for linear recurrences with constant coefficients, signature (3,1).

Cf. A003688, A052906.
Appears in A180142.
Cf. A000602 (more information on n-alkanes).

a:= n-> (<<0|1>, <1|3>>^n. <<2, 5>>)[1,1]:
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 1}, {5, 7}, 20] (* Eric W. Weisstein, Jul 14 2021 *)
CoefficientList[Series[(2 - x)/(1 - 3 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 14 2021 *)

a(n)=([0,1;1,3]^n*[2;5])[1,1] \\ Charles R Greathouse IV, Oct 13 2016

G.f.: (2-x)/(1-3*x-x^2).
a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
a(n) = ((4+7*A)*A^(-n-1) + (4+7*B)*B^(-n-1))/13 with A = (-3+sqrt(13))/2 and B = (-3-sqrt(13))/2.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^n*2/(A006497(n) - A006190(n)*sqrt(13)).
a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 5 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024
a(n) = 2*A006190(n+1) - A006190(n). - R. J. Mathar, Feb 14 2024

A002094 Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.

Original entry on oeis.org

0, 1, 2, 5, 10, 25, 56, 139, 338, 852, 2145, 5513, 14196, 36962, 96641, 254279, 671640, 1781840, 4742295, 12662282, 33898923, 90981264, 244720490, 659591378, 1781048728, 4817420360, 13050525328, 35405239155, 96180222540, 261603173201, 712364210543

Offset: 1

N. J. A. Sloane

nonn

A pair of parallel edges is permitted and is regarded as a cycle of length 2.
The original definition in A Handbook of Integer Sequences (1973) based on Schiff (1875) was simply "Alcohols". - N. J. A. Sloane, Mar 22 2018
Schiff used an now outdated terminology and did not use the term "alcohols", but in German "zweiwerthige Kohlenwasserstoffe C_{n}H_{2n} ..." and later "... deren je zwei verfuegbare Affinitaeten ... durch Alkoholradikale befriedigt sind.", translated "bivalent hydrocarbons ... whose free valences ... are covered by alcohol radicals". At that time the meaning of "alcohol radical" was different from modern terminology, now meaning an -OH group, but in Schiff's terminology another C_{x}H{y} hydrocarbon group was meant. The organic compounds that are described by the graphs of this sequence in modern chemical terminology are the acyclic alkenes, with exactly one double carbon-to-carbon bond, and the monocyclic cycloalkanes (see Wikipedia links). - Hugo Pfoertner, Mar 29 2018

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..400
R. J. Mathar, Illustration for graphs up to 6 carbons, 2018
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
Wikipedia, Alkene. Those with exactly one double carbon-to-carbon bond are covered by this sequence, the simplest being ethylene C_{2}H_{4}.
Wikipedia, Cycloalkane. The simplest alicyclic compounds, which are the monocyclic saturated hydrocarbons with formula C_{n}H_{2n}, are covered by this sequence, the first example being cyclopropane C_{3}H_{6}.

Cf. A000294, A000598, A000602, A000625, A000642, A001429 (unbound degrees), A068051.

# cycle index of cyclic group C_n
cycC_n := proc(n::integer,a)
    local d ;
    add(numtheory[phi](d)*a[d]^(n/d),d=numtheory[divisors](n)) ;
    %/n ;
end proc:
# cycle index of dihedral group
cyD_n := proc(n::integer,a)
    cycC_n(n,a)/2 ;
    if type(n,'odd') then
        %+ a[1]*a[2]^((n-1)/2)/2 ;
    else
        %+ ( a[1]^2*a[2]^((n-2)/2) +a[2]^(n/2) )/4 ;
    end if;
end proc:
a000642 := [
    1, 1, 2, 3, 7, 14, 32, 72, 171, 405, 989, 2426, 6045, 15167, 38422, 97925,
    251275, 648061, 1679869, 4372872, 11428365, 29972078, 78859809, 208094977,
    550603722, 1460457242, 3882682803, 10344102122, 27612603765, 73844151259,
    197818389539, 530775701520, 1426284383289] ;
g := [add(a000642[i]*x^i,i=1..nops(a000642)) ];
for i from 2 to nops(a000642) do
    g := [op(g), subs(x=x^i,g[1]) ] ;
end do:
Nmax := nops(a000642) :
G := 0 ;
for c from 2 to Nmax do
    f := cyD_n(c,g) ;
    G := G+ taylor(f,x=0,Nmax) ;
end do:
taylor(G,x=0,Nmax) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 17 2018

terms = 31;
cycC[n_, a_] := Sum[EulerPhi[d] a[[d]]^(n/d), {d, Divisors[n]}]/n;
cyD[n_, a_] := Module[{cc}, cc = (1/2)cycC[n, a]; If[OddQ[n], (1/2)a[[1]]* a[[2]]^((n-1)/2)+cc, (1/4)(a[[1]]^2 a[[2]]^((n-2)/2) + a[[2]]^(n/2)) + cc]];
B[] = 0; Do[B[x] = Normal[(1/6) x (2 B[x^3] + 3 B[x^2] B[x] + B[x]^3) + O[x]^terms+1], terms];
A[x_] = (1/2) x (B[x^2] + B[x]^2) + O[x]^(terms+2);
a000642 = Rest[CoefficientList[A[x], x]];
g = {Sum[x^i a000642[[i]], {i, 1, terms+1}]};
For[i = 2, i <= Length[a000642], i++, g = Flatten[Append[g, g[[1]] /. x -> x^i]]];
For[G = 0; c = 2, c < terms+1, c++, f = cyD[c, g]; G = Series[f, {x, 0, terms+1}] + G];
Most[Rest[CoefficientList[G, x]]] (* Jean-François Alcover, Mar 26 2020, after R. J. Mathar *)

Let A(x) denote the generating function for A000598 (Number of rooted ternary trees with n nodes), i.e., A(x) = 1+(1/6)*x*(A(x)^3+3*A(x)*A(x^2)+2*A(x^3)), and set B(x)=(A(x)^2+A(x^2))/2. With D_k(x) being the cycle polynomial of the regular k-gon for k>=2, the final generating function is then given by Sum_{k>=2} x^k*D_k(B(x)), which can be evaluated very quickly. - Sascha Kurz, Mar 18 2018

Better definition from R. J. Mathar; terms beyond 852 from R. J. Mathar and Sean A. Irvine, Mar 17 2018

A010372 Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) with a centroid ignoring stereoisomers.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 9, 8, 35, 39, 159, 202, 802, 1078, 4347, 6354, 24894, 38157, 148284, 237541, 910726, 1511717, 5731580, 9816092, 36797588, 64658432, 240215803, 431987953, 1590507121, 2917928218, 10660307791, 19910436898

Offset: 1

Paul Zimmermann, N. J. A. Sloane

The degree of each node is <= 4.
A centroid is a node with less than n/2 nodes in each of the incident subtrees, where n is the number of nodes in the tree. If a centroid exists it is unique.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A086194 for the analogous sequence with stereoisomers counted.

F. Harary, Graph Theory, p. 36, for definition of centroid.

A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
Index entries for sequences related to trees

Cf. A010373, A000022, A086194, A000598, A000602.

A000602(n) = a(n) + A010373(n/2) for n even, A000602(n) = a(n) for n odd.

with(combstruct): Alkyl := proc(n) combstruct[count]([ U,{U=Prod(Z,Set(U,card<=3))},unlabeled ],size=n) end:
centeredHC := proc(n) option remember; local f,k,z,f2,f3,f4; f := 1 + add(Alkyl(k)*z^k, k=0..iquo(n-1,2));
f2 := series(subs(z=z^2,f), z, n+1); f3 := series(subs(z=z^3,f), z, n+1); f4 := series(subs(z=z^4,f), z, n+1);
f := series(f*f3/3+f4/4+f2^2/8+f2*f^2/4+f^4/24, z, n+1); coeff(f, z, n-1) end: seq(centeredHC(n), n=1..32);

Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A010373 Number of unrooted quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n+2) with a bicentroid, ignoring stereoisomers.

Original entry on oeis.org

1, 1, 3, 10, 36, 153, 780, 4005, 22366, 128778, 766941, 4674153, 29180980, 185117661, 1193918545, 7800816871, 51584238201, 344632209090, 2324190638055, 15804057614995, 108277583483391, 746878494484128, 5183852459907628

Offset: 1

Paul Zimmermann, N. J. A. Sloane

The degree of each node is <= 4.
A bicentroid is an edge which connects two subtrees of exactly m/2 nodes, where m is the number of nodes in the tree. If a bicentroid exists it is unique. Clearly trees with an odd number of nodes cannot have a bicentroid.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A086200 for the analogous sequence with stereoisomers counted.

F. Harary, Graph Theory, p. 36, for definition of bicentroid.

Alois P. Heinz, Table of n, a(n) for n = 1..500 (terms n = 1..100 from Vincenzo Librandi)
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
Index entries for sequences related to trees

A000602(n) = A010372(n) + a(n/2) for n even, A000602(n) = A010372(n) for n odd.

Cf. A000200, A000598.

M[1146] := [ T,{T=Union(Epsilon,U),U=Prod(Z,Set(U,card<=3))},unlabeled ]:
bicenteredHC := proc(n) option remember; if n mod 2<>0 then 0 else binomial(count(M[ 1146 ],size=n/2)+1,2) fi end:

m = 24; a[x_] = Sum[c[k]*x^k, {k, 0, m}]; s[x_] = Series[ 1 + (1/6)*x*(a[x]^3 + 3*a[x]*a[x^2] + 2*a[x^3]) - a[x], {x, 0, m}]; eq = Thread[ CoefficientList[s[x], x] == 0];
Do[so[k] = Solve[eq[[1]], c[k-1]][[1]]; eq = Rest[eq] /. so[k], {k, 1, m+1}]; b = Array[c, m, 0] /. Flatten[ Array[so, m+1] ]; Rest[b*(b+1)/2] (* Jean-François Alcover, Jul 25 2011, after A000598 *)

a(n) = b(n)*(b(n)+1)/2, where b(n) = A000598[ n ].

Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A112408 Number of connected simple graphs with n vertices, n+3 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 2, 14, 79, 430, 2161, 10162, 45282, 192945, 790849, 3138808, 12116550, 45675153, 168661704, 611701138, 2183635232, 7686541342, 26720976964, 91856241351, 312594121721, 1054104924270

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 4 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, this sequence, A112424, A112425, A112426, A112442. Cf. A121941.

for n in {5..15}; do geng -c -D4 ${n} $((n+3)):$((n+3)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset, new name, and a(18) from Andrey Zabolotskiy, Nov 24 2017

a(18)-a(24) added by Georg Grasegger, Jun 05 2023

A112424 Number of connected simple graphs with n vertices, n+4 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 59, 427, 2768, 16461, 90111, 460699, 2222549, 10216607, 45076266, 192059940, 794088479, 3198709835, 12593964702, 48596474890, 184195614359, 687087962550, 2526421534903

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 5 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, A112408, this sequence, A112425, A112426, A112442. Cf. A121941.

for n in {5..15}; do geng -c -D4 ${n} $((n+4)):$((n+4)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(16)-a(23) added by Georg Grasegger, Jun 05 2023

A112425 Number of connected simple graphs with n vertices, n+5 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 31, 298, 2616, 20346, 140605, 880737, 5082279, 27402524, 139587885, 677772953, 3158930531, 14212444473, 62009204208, 263350765116, 1092085621098, 4433596269478

Offset: 1

Jonathan Vos Post, Dec 21 2005

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 6 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. [gives different a(15)]

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, A112408, A112424, this sequence, A112426, A112442. Cf. A121941.

for n in {5..13}; do geng -c -D4 ${n} $((n+5)):$((n+5)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(15) corrected and a(16)-a(22) added by Georg Grasegger, Jun 05 2023

A112426 Number of connected simple graphs with n vertices, n+6 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 9, 134, 1714, 18436, 167703, 1327240, 9372119, 60324933, 359730035, 2012733260, 10670975762, 54028108819, 262872075003, 1235323112178, 5630370812614

Offset: 1

Jonathan Vos Post, Dec 21 2005

Distribution of carbon skeletons. See the paper by Hendrikson and Parks for details. If n=6 the number of 7-cyclic skeletons is 1. If n=7 the number of 7-cyclic skeletons is 9. If n=10 the number of 7-cyclic skeletons is 18436. - Parthasarathy Nambi, Jan 05 2007

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 7 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs [gives different numbers for n > 10].

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, A112408, A112424, A112425, this sequence, A112442. Cf. A121941.

for n in {6..13}; do geng -c -D4 ${n} $((n+6)):$((n+6)) -u; done # Andrey Zabolotskiy, Nov 24 2017

New name, offset corrected, and a(11)-a(14) corrected by Andrey Zabolotskiy, Nov 24 2017

a(15)-a(21) added by Georg Grasegger, Jun 05 2023

cp's OEIS Frontend

A000200 Number of bicentered hydrocarbons with n atoms.

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A112410 Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.

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nauty

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A180148 a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.

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A002094 Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.

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A010372 Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) with a centroid ignoring stereoisomers.

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A010373 Number of unrooted quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n+2) with a bicentroid, ignoring stereoisomers.

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A112408 Number of connected simple graphs with n vertices, n+3 edges, and vertex degrees no more than 4.

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