A007869 -id:A007869 - cp's OEIS frontend

A007970 Rhombic numbers.

3, 7, 8, 11, 15, 19, 23, 24, 27, 31, 32, 35, 40, 43, 47, 48, 51, 59, 63, 67, 71, 75, 79, 80, 83, 87, 88, 91, 96, 99, 103, 104, 107, 115, 119, 120, 123, 127, 128, 131, 135, 136, 139, 143, 151, 152, 159, 160, 163, 167, 168, 171, 175, 176, 179

Offset: 1

J. H. Conway

nonn

A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n)). - Reinhard Zumkeller, Jun 18 2011
This sequence gives the values d of the Pell equation x^2 - d*y^2 = +1 that have positive fundamental solutions (x0, y0) with odd y0. This was first conjectured and is proved provided Conway's theorem in the link is assumed and the proof of the conjecture stated in A007869, given there in a W. Lang link, is used. - Wolfdieter Lang, Sep 19 2015
For a proof of Conway's theorem on the happy number factorization see the W. Lang link (together with the link given under A007969). - Wolfdieter Lang, Oct 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..99
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Wolfdieter Lang, Proof of a Theorem Related to the Happy Number Factorization.

Every number belongs to exactly one of A000290, A007969, A007970.
Cf. A007968.
Subsequence of A000037, A002145 is a subsequence.
A263008 (T numbers), A263009 (U numbers).

a007970 n = a007970_list !! (n-1)
a007970_list = filter ((== 2) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015

r[b_, c_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + 2 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]);
f[n_] := f[n] = If[! IntegerQ[Sqrt[n]], Catch[Do[{b, c} = bc; If[ (r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]]; If[ (r0 = r[c, b]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]], {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]} ]]];
A007970 = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* Jean-François Alcover, Jun 26 2012 *)

a(n) = A191856(n)*A191857(n); A007968(a(n))=2. - Reinhard Zumkeller, Jun 18 2011

a(n) is in the sequence if a(n) = D*E with positive integers D and E, such that E*U^2 - D*T^2 = 2 has an integer solution with U*T odd (without loss of generality one may take U and T positive). See the Conway link. D and E are given in A191856 and A191857, respectively. - Wolfdieter Lang, Oct 05 2015

159 and 175 inserted by Jean-François Alcover, Jun 26 2012

A000171 Number of self-complementary graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 10, 36, 0, 0, 720, 5600, 0, 0, 703760, 11220000, 0, 0, 9168331776, 293293716992, 0, 0, 1601371799340544, 102484848265030656, 0, 0, 3837878966366932639744, 491247277315343649710080, 0, 0

Offset: 1

N. J. A. Sloane

a(n) = A007869(n)-A054960(n), where A007869(n) is number of unlabeled graphs with n nodes and an even number of edges and A054960(n) is number of unlabeled graphs with n nodes and an odd number of edges.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 139, Table 6.1.1.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
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 = 1..100
H. Fripertinger, Self-complementary graphs
Victoria Gatt, Mikhail Klin, Josef Lauri, Valery Liskovets, From Schur Rings to Constructive and Analytical Enumeration of Circulant Graphs with Prime-Cubed Number of Vertices, in Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, (Pilsen, Czechia, WAGT 2016) Vol. 305, Springer, Cham, 37-65.
Richard A. Gibbs, Self-complementary graphs J. Combinatorial Theory Ser. B 16 (1974), 106--123. MR0347686 (50 #188). - _N. J. A. Sloane_, Mar 27 2012
Sebastian Jeon, Tanya Khovanova, 3-Symmetric Graphs, arXiv:2003.03870 [math.CO], 2020.
B. D. McKay, Self-complementary graphs
R. C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc., 38 (1963), 99-104.
Eric Weisstein's World of Mathematics, Self-Complementary Graph
D. Wille, Enumeration of self-complementary structures, J. Comb. Theory B 25 (1978) 143-150

Cf. A047660, A051251, A047832.

Cf. A008406 (triangle of coefficients of the "graph polynomial").

< -1, {n, 1, 20}]  (* Geoffrey Critzer, Oct 21 2012 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
a[n_] := Module[{s = 0}, Switch[Mod[n, 4], 2|3, 0, _, Do[s += permcount[4 p]*2^edges[p]*If[OddQ[n], n*2^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]; s/n!]];
Array[a, 40] (* Jean-François Alcover, Aug 26 2019, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + 2*sum(i=1, #v, v[i])}
a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p)) * 2^edges(p) * if(n%2, n*2^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 16 2018

a(4n) = A003086(2n).

a(4*n+1) = A047832(n), a(4*n+2) = a(4*n+3) = 0. - Andrew Howroyd, Sep 16 2018

More terms from Ronald C. Read and Vladeta Jovovic

A054960 Number of unlabeled graphs with n nodes and an odd number of edges.

Original entry on oeis.org

0, 1, 2, 5, 16, 78, 522, 6168, 137316, 6002584, 509498932, 82545585936, 25251015681176, 14527077828617744, 15713242984902154384, 32000507852263778595584, 122967932076766466336249888, 893788862572805850273939095424, 12318904626562502262191503745716384

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000088, A000171.

Cf. A007869 for graphs with an even number of edges.

Table[Total[Table[NumberOfGraphs[n,m],{m,1,Binomial[n,2],2}]],{n,1,15}]  (* Geoffrey Critzer, Oct 20 2012 *)

a(n) = (A000088(n) - A000171(n))/2.

More terms from Vladeta Jovovic, Jul 19 2000

Terms a(18) and beyond from Andrew Howroyd, Sep 17 2018

A054928 Number of complementary pairs of directed graphs on n nodes. Also number of unlabeled digraphs with n nodes and an even number of arcs.

Original entry on oeis.org

1, 2, 10, 114, 4872, 770832, 441038832, 896679948304, 6513978501814144, 170630215981070456064, 16261454692532635025585792, 5683372715412701087902846672384, 7334542846356464937798016155801130496, 35157828307617499760694672217473135511928832

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000273, A003086, A007869, A054960, A055969.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
b[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
edges4[v_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 v[[i]] - 1, {i, 1, Length[v]}];
c[n_] := (s = 0; Do[s += permcount[2 p]*2^edges4[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);
a[n_] := (b[n] + c[n])/2;
Array[a, 14] (* Jean-François Alcover, Aug 26 2019, using Andrew Howroyd's code for b=A000273 and c=A003086 *)

Average of A000273 and A003086.

More terms from Vladeta Jovovic, Jul 19 2000

Terms a(14) and beyond from Andrew Howroyd, Sep 17 2018

A055969 Number of unlabeled digraphs with n nodes and an odd number of arcs.

Original entry on oeis.org

0, 1, 6, 104, 4736, 770112, 440994608, 896679244544, 6513978322585408, 170630215971902124288, 16261454692523251085611648, 5683372715412699486531047331840, 7334542846356464931239079919515090432, 35157828307617499760690834338506768579289088

Offset: 1

Vladeta Jovovic, Jul 19 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A054928, A007869, A054960.

A000273 = Cases[Import["https://oeis.org/A000273/b000273.txt", "Table"], {, }][[All, 2]];
A003086 = Cases[Import["https://oeis.org/A003086/b003086.txt", "Table"], {, }][[All, 2]];
a[n_] := (A000273[[n + 1]] - A003086[[n]])/2;
Array[a, 50] (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd *)

a(n) = (A000273(n)-A003086(n))/2.

Terms a(14) and beyond from Andrew Howroyd, Sep 17 2018

A055973 Number of unlabeled digraphs with loops (relations) on n nodes and with an even number of arcs.

Original entry on oeis.org

1, 1, 6, 52, 1540, 145984, 48467296, 56141454464, 229148555640864, 3333310786076963968, 174695272746896566439424, 33301710992539090379269318144, 23278728241293494584257987458067456, 60084295633556503802059558812644803074048, 576025077880237078776946976495257043823396069376

Offset: 0

Vladeta Jovovic, Jul 19 2000

Also relations considered equivalent when isomorphic and when complemented. - Christian G. Bower, Jan 05 2004

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000171, A000595, A007869, A047832, A054928, A055974.

a(2*n) = (A000595(2*n) + A047832(n))/2; a(2*n+1) = A000595(2*n+1)/2.
a(n) = (A000595(n) + A000171(2*n+1))/2.
a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2=2} (fixA[s_1, s_2, ...;t_1, t_2]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!)) where fixA[...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*t_d))^(gcd(i, j)*s_i*s_j)) - Christian G. Bower, Jan 05 2004

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, Apr 19 2020

A230367 Number of colorings of the edges of the complete graph on n unlabeled vertices using at most three interchangeable colors under the symmetries of the full edge permutation group.

Original entry on oeis.org

1, 3, 15, 142, 4300, 384199, 98654374, 70130880569, 136638863494089, 730439999032117301, 10764688922047900738650, 439762062635963206090747374, 50066701349010686289507943943535, 15962815411172611585301863116082363362, 14314975828662356561039590680011420432741442, 36247244119877673111912410070361564495415461430358

Offset: 2

Marko Riedel, Dec 20 2013

nonn

Harary and Palmer, Graphical Enumeration, Chapter Six.

Marko Riedel, Table of n, a(n) for n = 2..40
Marko Riedel, Colorings of the cube.
Marko Riedel, Colorings of the complete graph Kn with some number of swappable colors.

Cf. A007869.

A233748 Number of colorings of the edges of the complete graph on n unlabeled vertices using at most four colors up to permutation of the colors.

Original entry on oeis.org

1, 3, 22, 513, 67685, 37205801, 74992370359, 543437207831908, 14224090440652751128, 1355263603548588163939892, 473629305416562052216063601182, 611217587951920565260750675133929534, 2929826971690847578421964010597063713803102, 52431093582529289457516130655642218612938867242516

Offset: 2

Marko Riedel, Dec 20 2013

nonn

Harary and Palmer, Graphical Enumeration, Chapter Six.

Marko Riedel, Table of n, a(n) for n = 2..40
Marko Riedel, Colorings of the cube.
Marko Riedel, Colorings of the complete graph Kn with some number of swappable colors.

Cf. A007869.

Definition clarified by Andrew Howroyd, Feb 04 2024

A233894 Number of colorings of the edges of the complete graph on n unlabeled vertices using at most five interchangeable colors under the symmetries of the full edge permutation group.

Original entry on oeis.org

1, 3, 24, 956, 370438, 794610689, 7713545142724, 334331083961076765, 65276369289196100770910, 57946272822007395195661933018, 235778940244496842474046899073666629, 4427916979407079482171816367439500693634123, 386083248154389925434229068715450099160359622470252

Offset: 2

Marko Riedel, Dec 20 2013

nonn

Harary and Palmer, Graphical Enumeration, Chapter Six.

Marko Riedel, Table of n, a(n) for n = 2..40
Marko Riedel, Colorings of the cube.
Marko Riedel, Colorings of the complete graph Kn with some number of swappable colors.

Cf. A007869.

More terms from Marko Riedel Feb 04 2024

A309116 a(n) = number of cographs on n points.

Original entry on oeis.org

1, 3, 25, 1299, 1974452, 94345468975, 152799292695935115, 10526127565809458484649781, 38375912431199015810067477044326371, 9002076475560099357419498216602893054297145089, 162015966626938926212463690033352243299416773774432388589099

Offset: 2

Michael De Vlieger, Jul 13 2019

nonn

Here, a cograph is basically a partition of unlabeled edges of the complete graph on n unlabeled vertices. - Andrey Zabolotskiy, Aug 27 2022

Robert Haas, Cographs, arXiv:1905.12627 [math.GM], 2019, p. 3, 57.
Robert Haas, Intersection Cographs and Aesthetics, Journal of Humanistic Mathematics, 12 (2022), 4-23.
Marko Riedel, Colorings of the complete graph Kn with any number of swappable colors.

Cf. partitions into no more than 2..5 parts: A007869, A230367, A233748, A233894.

cycleIndSymm[n_] := cycleIndSymm[n] = CoefficientRules[CycleIndexPolynomial[ SymmetricGroup[n], x /@ Range[n]], x /@ Range[n]];
cycleIndEdge[n_] := cycleIndEdge[n] = CoefficientRules[Sum[Last[t] With[{tt = First[t]}, With[{ind = Flatten@Position[tt, Except[0], Heads -> False]}, Product[x[LCM@@p]^(GCD@@p Times@@tt[[p]]), {p, Subsets[ind, {2}]}] Product[With[{e = tt[[k]]}, x[k]^(k e (e-1)/2 + Quotient[k-1, 2] e) If[EvenQ[k], x[k/2]^e, 1]], {k, ind}]]], {t, cycleIndSymm[n]}], x /@ Range[n (n-1)/2]];
v[n_, m_] := With[{dv = Divisors /@ Range[m]}, Sum[Last[a] With[{ra = Flatten@Position[First@a, Except[0], Heads -> False]}, Sum[Last[b] Product[(dv[[va]].b[[1, dv[[va]]]])^a[[1, va]], {va, ra}], {b, cycleIndSymm[m]}]], {a, cycleIndEdge[n]}]];
a[2] = 1; a[3] = 3;
a[n_] := 1 + v[n, -1 + n (n-1)/2];
Table[a[n], {n, 2, 7}] (* Andrey Zabolotskiy, Feb 06 2024, after Marko Riedel *)

a(6)-a(9) from Andrey Zabolotskiy, Aug 27 2022
a(10) from Andrey Zabolotskiy, Feb 06 2024
a(11)-a(12) from Andrey Zabolotskiy, Feb 26 2025

cp's OEIS Frontend

A007970 Rhombic numbers.

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A000171 Number of self-complementary graphs with n nodes.

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A054960 Number of unlabeled graphs with n nodes and an odd number of edges.

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A054928 Number of complementary pairs of directed graphs on n nodes. Also number of unlabeled digraphs with n nodes and an even number of arcs.

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A055969 Number of unlabeled digraphs with n nodes and an odd number of arcs.

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A055973 Number of unlabeled digraphs with loops (relations) on n nodes and with an even number of arcs.

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A230367 Number of colorings of the edges of the complete graph on n unlabeled vertices using at most three interchangeable colors under the symmetries of the full edge permutation group.

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A233748 Number of colorings of the edges of the complete graph on n unlabeled vertices using at most four colors up to permutation of the colors.

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