cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A125670 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 1. See the paper by Hendrickson and Parks for details.

Original entry on oeis.org

1, 2, 9, 26, 87, 257, 787, 2322, 6891, 20160, 58939, 171203, 496294, 1433558, 4132744, 11886827, 34133563, 97856500, 280172582, 801174478, 2288600128, 6531205571, 18622839635, 53059229091, 151067980960, 429840337630, 1222335365450, 3474107883033, 9869276762717
Offset: 5

Views

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

Comments

Here 'alpha' is the number of atoms the two rings have in common.
Equivalently, the number of graphs on n unlabeled nodes with exactly 2 cycles joined at a single node and all nodes having degree at most 4. See A121158 for the special case of both cycles having the same length. - Andrew Howroyd, May 24 2018

Examples

			If n=5 then the number of bicyclics when 'alpha' = one is 1.
If n=6 then the number of bicyclics when 'alpha' = one is 2.
If n=7 then the number of bicyclics when 'alpha' = one is 9.
If n=8 then the number of bicyclics when 'alpha' = one is 26.
From _Andrew Howroyd_, May 24 2018: (Start)
Case n = 6: the two cases are a 3-cycle joined to a 4-cycle and a 3-cycle joined to another 3-cycle with a pendant edge.
   o---o-----o     o---o---o
    \ / \    |      \ / \ /
     o   o---o       o   o---o
(End)

References

  • James B. Hendrickson and Camden A. Parks, "Generation and Enumeration of Carbon Skeletons", J. Chem. Inf. Comput. Sci., vol. 31 (1991), pp. 101-107. See Table VII column 3 on page 104.

Links

Crossrefs

Cf. A121158, A121330, A121331, A121332, A121333, A125064.

Programs

  • PARI
    \\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
CycleIndex(n)={(sum(i=1, (n-1)\2-1, sum(j=1, (n-1)\2-i, (j1^(2*(i+j)) + 2*j1^(2*i)*j2^j + j2^(i+j))*(1 + j1)^2)) + sum(k=1, (n-1)\4,  2*(j2^(2*k) + j4^k)*(1 + j2)))/8}
seq(n)={my(t=G(n)); t=x*(t^2+subst(t, x, x^2))/2; my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x^n)); Vec(substvec(CycleIndex(n), [j1,j2,j4], [g(1),g(2),g(4)]))} \\ Andrew Howroyd, May 24 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, May 24 2018

A125669 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 0. See the paper by Hendrickson and Parks for details.

Original entry on oeis.org

1, 4, 20, 76, 288, 1005, 3433, 11324, 36712, 116809, 367076, 1140226, 3510491, 10722708, 32539364, 98178211, 294767639, 881147521, 2623934079, 7787024985, 23039064263, 67977412951, 200072442611, 587532484513, 1721812143140, 5036454320102, 14706743476128
Offset: 6

Views

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

Comments

Here 'alpha' is the number of atoms the two rings have in common.
Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles without any shared node and all nodes having degree at most 4. See A121162 for the special case of the two cycles having the same length. - Andrew Howroyd, May 25 2018

Examples

			If n=6 then the number of bicyclics when 'alpha' = zero is 1.
If n=7 then the number of bicyclics when 'alpha' = zero is 4.
If n=8 then the number of bicyclics when 'alpha' = zero is 20.
If n=9 then the number of bicyclics when 'alpha' = zero is 76.
From _Andrew Howroyd_, May 25 2018: (Start)
Case n=7: illustrations of the 4 graphs:
     o   o   o       o   o   o       o   o---o       o   o---o
    / \ / \ / \     / \ /   / \     / \     / \     / \   \   \
   o---o   o---o   o---o---o---o   o---o---o---o   o---o---o---o
(End)

Links

Crossrefs

Cf. A121162, A121330, A121331, A121332, A121333, A125064, A125670.

Programs

  • PARI
    \\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={sum(i=1, n\2-1, sum(j=1, n\2-i, (d1^(2*(i+j)) + 2*d1^(2*i)*d2^j + d2^(i+j))*(1 + d1)^2))/(8*(1-d1))}
C2(n)={sum(k=1, n\4,  2*(d2^(2*k) + d4^k)*(1 + d2))*(1+d1)/(8*(1-d2))}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2],[g(d,1),g(d,2)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d4], [g(d,1),g(d,2),g(d,4)]))} \\ Andrew Howroyd, May 25 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, May 25 2018

A125671 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 2. See the paper by Hendrickson and Parks for details.

Original entry on oeis.org

1, 3, 11, 32, 100, 294, 881, 2590, 7639, 22344, 65278, 189832, 550846, 1593558, 4600435, 13251623, 38104280, 109382300, 313543725, 897588156, 2566575323, 7331196543, 20921299025, 59653124923, 169959192844, 483897197563, 1376848221698, 3915320424705, 11128029239672
Offset: 4

Views

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

Comments

Here 'alpha' is the number of atoms the two rings have in common.
Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles joined at a single edge and all nodes having degree at most 4. See A121165 for the special case of the two cycles having the same length. - Andrew Howroyd, May 25 2018

Examples

			If n=5 then the number of bicyclics when 'alpha' = two is 3.
If n=6 then the number of bicyclics when 'alpha' = two is 11.
If n=7 then the number of bicyclics when 'alpha' = two is 32.
If n=8 then the number of bicyclics when 'alpha' = two is 100.
Case n=5: illustration of the 3 graphs:
.
       o---o        o---o       o   o
      /|   |       /|\         /|\  |
     / |   |      / | \       / | \ |
    /  |   |     /  |  \     /  |  \|
   o---o---o    o---o---o   o---o---o

Links

Crossrefs

Cf. A121165, A121330, A121331, A121332, A121333, A125064, A125670.

Programs

  • PARI
    \\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={(sum(k=2, n, (k-1)*d1^k) + sum(k=1, n\2, d2^k))/4}
C2(n)={(sum(k=1, n\2, d2^k) + sum(i=1, n-1, sum(j=1, n-i, d2^(i\2+j\2) * d1^(i%2+j%2))))/4}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2],[g(d,1),g(d,2)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d4], [g(d,1),g(d,2),g(d,4)]))} \\ Andrew Howroyd, May 25 2018

Extensions

a(4) prepended and terms a(16) and beyond from Andrew Howroyd, May 25 2018

A261340 Decimal expansion of the radius of convergence of the generating function of A000598, the number of rooted ternary trees of n vertices.

Original entry on oeis.org

3, 5, 5, 1, 8, 1, 7, 4, 2, 3, 1, 4, 3, 7, 7, 3, 9, 2, 8, 8, 2, 2, 4, 4, 4, 7, 3, 6, 4, 7, 6, 3, 2, 6, 3, 6, 7, 0, 8, 7, 4, 6, 9, 5, 4, 1, 7, 5, 3, 2, 2, 1, 3, 4, 2, 3, 8, 1, 2, 9, 4, 9, 9, 7, 1, 2, 8, 0, 0, 1, 8, 0, 5, 7, 5, 5, 5, 7, 8, 2, 8, 8, 6, 7, 9, 8, 1, 3, 8, 1, 0, 8, 2, 4, 1, 6, 7
Offset: 0

Views

Author

Jean-François Alcover, Aug 15 2015

Keywords

Examples

			0.35518174231437739288224447364763263670874695417532...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 298.

Links

Crossrefs

Cf. A000598, A000642, A002094, A121331, A244399.

Programs

  • Mathematica
    digits = 97; m = 2 digits + 10; For[gf = 0; i = 0, i <= m, i++, gf = Series[1 + x*(gf^3/6 + (gf /. x -> x^2)*gf/2 + (gf /. x -> x^3)/3), {x, 0, m + 1}] // Normal];
g[r_] := Module[{r2, r3, X, ym}, r2 = gf /. x -> r^2; r3 = gf /. x -> r^3; X[y_] = (y - 1)/(y^3/6 + r2*y/2 + r3/3); ym = y /. FindRoot[X'[y] == 0, {y, 2}, WorkingPrecision -> digits + 5]; X[ym]]; rho = FixedPoint[g, 1/3, SameTest -> (Abs[#1 - #2] < 10^-digits &)]; RealDigits[rho, 10, digits] // First

Extensions

More digits from Vaclav Kotesovec, Aug 15 2015
More digits and Mma code updated by Jean-François Alcover, Apr 18 2016

A305132 Number of connected graphs on n unlabeled nodes with exactly 2 cycles joined along two or more edges but not more than half each cycle and all nodes having degree at most 4.

Original entry on oeis.org

1, 3, 11, 36, 116, 366, 1151, 3583, 11093, 34141, 104489, 318139, 963899, 2907276, 8731919, 26125538, 77889504, 231466147, 685811867, 2026481941, 5973064855, 17565416721, 51547293439, 150977445294, 441409701444, 1288409915625, 3754926609800, 10927779696264
Offset: 5

Views

Author

Andrew Howroyd, May 26 2018

Keywords

Comments

The resulting graph will actually have three cycles. See A121331 for the special case of all three cycles having the same length.
Equivalently, the number of connected simple graphs with n unlabeled nodes and n + 1 edges and all nodes having degree at most 4 (A112410) less those graphs described by A125669, A125670 and A125671.

Examples

			Illustration of graphs for n=5 and n=6:
    o          o--o       o           o--o
   /|\        /|\        /|\         /|  |
  o o o      o o o      o o o--o    o o  |
   \|/        \|/        \|/         \|  |
    o          o          o           o--o

Links

Crossrefs

Cf. A112410, A121331, A125669, A125670, A125671, A125672, A125673.

Programs

  • PARI
    \\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={subst(Pol(x^3*d1^3/(1-x*d1)^3 + 3*x^3*d1*d2/((1-x*d1)*(1-x^2*d2)) + 2*x^3*d3/(1-x^3*d3) + O(x*x^n)), x, 1)/12}
C2(n)={subst(Pol(((x*d1+x^2*d2)/(1-x^2*d2))^3 + 3*(x*d1+x^2*d2)*x^2*d2/(1-x^2*d2)^2 + 2*(x^3*d3 + x^6*d6)/(1-x^6*d6) + O(x*x^n)), x, 1)/12}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2,d3],[g(d,1), g(d,2), g(d,3)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d3,d6], [g(d,1), g(d,2), g(d,3), g(d,6)]))}

Formula

a(n) >= A125672(n) + A125673(n).

A125672 Classification of bicyclics with the parameter 'alpha' having the value of 3. See the paper by Hendrickson and Parks for details.

Original entry on oeis.org

1, 3, 11, 35, 113, 351, 1093, 3348, 10218, 30906, 93005
Offset: 5

Views

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

Comments

Here 'alpha' is the number of atoms the two rings have in common.

Examples

			If n=5 then the number of bicyclics when 'alpha' = three is 1.
If n=6 then the number of bicyclics when 'alpha' = three is 3.
If n=7 then the number of bicyclics when 'alpha' = three is 11.
If n=8 then the number of bicyclics when 'alpha' = three is 35.

References

  • James B. Hendrickson and Camden A. Parks, "Generation and Enumeration of Carbon Skeletons", J. Chem. Inf. Comput. Sci., vol. 31 (1991), pp. 101-107. See Table VII column 5 on page 104.

Crossrefs

Cf. A121330, A121331, A121332, A121333, A125064.

A125673 Classification of bicyclics with the parameter 'alpha' having the value of 4. See the paper by Hendrickson and Parks for details.

Original entry on oeis.org

1, 3, 15, 57, 231, 853, 3131, 11007
Offset: 8

Views

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

Comments

Here 'alpha' is the number of atoms the two rings have in common.

Examples

			If n=8 then the number of bicyclics when 'alpha' = four is 1.
If n=9 then the number of bicyclics when 'alpha' = four is 3.
If n=10 then the number of bicyclics when 'alpha' = four is 15.
If n=11 then the number of bicyclics when 'alpha' = four is 57.

References

  • James B. Hendrickson and Camden A. Parks, "Generation and Enumeration of Carbon Skeletons", J. Chem. Inf. Comput. Sci., vol. 31 (1991), pp. 101-107. See Table VII column 6 on page 104.

Crossrefs

Cf. A121330, A121331, A121332, A121333, A125064.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.