A063832
Number of structurally isomeric homologs with molecular formula C_{3+n} H_{6+2n}.
Original entry on oeis.org
1, 1, 3, 6, 15, 33, 83, 196, 491, 1214, 3068, 7754, 19834, 50872, 131423, 340763, 887839, 2321193, 6090979, 16031341, 42319223, 112003765, 297164610, 790190726, 2105607907, 5621642203, 15036126167, 40284850520, 108102408101
Offset: 0
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
- G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987, p. 63.
- Ching-Wan Lam, "Enumeration of isomers of alkylcyclopropanes by means of alkyl 1,1-biradicals", J. Math. Chem., 27 (2000), 23-25. [From Parthasarathy Nambi, Aug 24 2008]
-
G[n_] := Module[{g}, Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
T[n_, k_] := Module[{t = G[n], g}, t = x*((t^2 + (t /. x -> x^2))/2); g[e_] = (Normal[t + O[x]^Quotient[n, e]] /. x -> x^e) + O[x]^n // Normal; Coefficient[(Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[ k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2])/2, x, n]];
a[n_] := T[n + 3, 3];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)
A116719
Number of monocyclic skeletons with n carbon atoms and a ring size of 4.
Original entry on oeis.org
1, 1, 4, 8, 24, 55, 147, 365, 954, 2431, 6327, 16369, 42743, 111595, 292849, 769805, 2030456, 5366844, 14222475, 37768154, 100510364, 267987501, 715847932, 1915406263, 5133382014, 13778469949, 37035674682, 99683747508, 268647638770, 724879674667, 1958151665752
Offset: 4
If n=5 then the number of monocyclic skeletons with ring size of four is 1.
-
G[n_] := Module[{g}, Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
T[n_, k_] := Module[{t = G[n], g}, t = x*((t^2 + (t /. x -> x^2))/2); g[e_] = (Normal[t + O[x]^Quotient[n, e]] /. x -> x^e) + O[x]^n // Normal; Coefficient[(Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[ k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2])/2, x, n]];
a[n_] := T[n, 4];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)
a(5) corrected and terms a(26) and beyond from
Andrew Howroyd, May 24 2018
A120333
Number of monocyclic skeletons with n carbon atoms and a ring size of 5.
Original entry on oeis.org
1, 1, 4, 9, 28, 71, 198, 521, 1418, 3773, 10153, 27114, 72705, 194531, 521447, 1397482, 3749836, 10067417, 27057233, 72779710, 195963184, 528127752, 1424707167, 3846943003, 10397057771, 28125235102, 76149287981, 206351312858, 559642013499, 1519019192097
Offset: 5
If n=10 then the number of monocyclic skeletons with ring size of five is 71.
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
-
G[n_] := Module[{g}, Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
T[n_, k_] := Module[{t = G[n], g}, t = x*((t^2 + (t /. x -> x^2))/2); g[e_] = (Normal[t + O[x]^Quotient[n, e]] /. x -> x^e) + O[x]^n // Normal; Coefficient[(Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[ k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2])/2, x, n]];
a[n_] := T[n + 5, 5];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)
A120779
Number of monocyclic skeletons with n carbon atoms and a ring size of 6.
Original entry on oeis.org
1, 1, 5, 12, 40, 106, 317, 868, 2462, 6778, 18801, 51561, 141583, 386865, 1056815, 2880894, 7850318, 21373466, 58182244, 158342918, 430954506, 1173001715, 3193466631, 8696388412, 23689689323, 64555962710, 175989086913, 479971320862, 1309581740793, 3574715516111
Offset: 6
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
A120790
Number of monocyclic skeletons with n carbon atoms and a ring size of 7.
Original entry on oeis.org
1, 1, 5, 13, 47, 136, 428, 1252, 3716, 10708, 30823, 87504, 247438, 694588, 1942866, 5411640, 15034045, 41659417, 115231598, 318231047, 877817312, 2419033683, 6661335192, 18332784221, 50432439047, 138692804603, 381332833730, 1048322091176, 2881742370070
Offset: 7
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
A120795
Number of monocyclic skeletons with n carbon atoms and a ring size of 8.
Original entry on oeis.org
1, 1, 6, 16, 63, 190, 631, 1923, 5940, 17706, 52573, 153442, 444935, 1277737, 3648896, 10357898, 29278749, 82435343, 231393069, 647760624, 1809395471, 5044728853, 14043361916, 39042161151, 108423265447, 300823420906, 834004924385, 2310725016923, 6398803186192
Offset: 8
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
A121156
Number of monocyclic skeletons with n carbon atoms and a ring size of 9.
Original entry on oeis.org
1, 1, 6, 18, 73, 240, 841, 2714, 8758, 27259, 83847, 252929, 754737, 2225129, 6505344, 18867696, 54384446, 155892016, 444839696, 1264336132, 3581584336, 10116577962, 28505061687, 80146304124, 224929999040, 630259883854, 1763574118096, 4928881367449
Offset: 9
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
A121157
Number of monocyclic skeletons with n carbon atoms and a ring size of 10.
Original entry on oeis.org
1, 1, 7, 21, 93, 319, 1170, 3931, 13168, 42365, 134372, 416765, 1275718, 3850137, 11501014, 34025545, 99891325, 291245242, 844287970, 2435145974, 6993264128, 20007298353, 57051831889, 162217819731, 460079126428, 1301975691316, 3677269460008, 10368105592318
Offset: 10
- Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
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