Georgi Guninski - cp's OEIS frontend

A182012 Number of graphs on 2n unlabeled nodes all having odd degree.

1, 3, 16, 243, 33120, 87723296, 3633057074584, 1967881448329407496, 13670271807937483065795200, 1232069666043220685614640133362240, 1464616584892951614637834432454928487321792, 23331378450474334173960358458324497256118170821672192, 5051222500253499871627935174024445724071241027782979567759187712

Offset: 1

Georgi Guninski, Apr 06 2012

As usual, "graph" means "simple graph, without self-loops or multiple edges".

The graphs on 2n vertices all having odd degrees are just the complements of those having all even degrees. That's why the property of all odd degrees is seldom mentioned. Therefore this sequence is just every second term of A002854. - Brendan McKay, Apr 08 2012

			The 3 graphs on 4 vertices are [(0, 3), (1, 3), (2, 3)], [(0, 2), (1, 3)], [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]: the tree with root of order 3, the disconnected graph consisting of two complete 2-vertex graphs, and the complete graph.

Sequence Fans Mailing List, Discussion, April 2012.
N. J. A. Sloane, The 16 graphs on 6 nodes

Cf. A210345, A210346, A000088. Bisection of A002854.

def graphsodddegree(MAXN=5):
    """
    requires optional package "nauty"
    """
    an=[]
    for n in range(1,MAXN+1):
        gn=graphs.nauty_geng("%s"%(2*n))
        cac={}
        a=0
        for G in gn:
            d = G.degree_sequence()
            if all(i % 2 for i in d):
                a += 1
        print('a(%s)=%s'%(n,a))
        an += [a]
    return an

a(n) = A002854(2n).

Terms from a(6) on added from A002854. - N. J. A. Sloane, Apr 08 2012

A171920 Numbers n with at least one solution to n=xyz, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

Original entry on oeis.org

1, 4, 9, 12, 16, 24, 25, 36, 40, 45, 49, 60, 64, 72, 81, 84, 100, 105, 112, 121, 144, 160, 169, 180, 189, 196, 216, 220, 225, 240, 256, 264, 280, 289, 297, 300, 312, 324, 352, 360, 361, 364, 385, 396, 400, 420, 429, 432, 441, 480, 484, 504, 520, 529, 544, 576

Offset: 1

Georgi Guninski, Oct 23 2010

nonn

Supersequence of A000290, i.e., all perfect squares are in the sequence.
Solutions (x,y) are integral points on the elliptic curve x*y*(x+y-1)=n. - Georgi Guninski, Oct 25 2010
From Robert G. Wilson v, Oct 25 2010: (Start)
a(n) != 2 (mod 3) nor {2, 3} (mod 4) nor 3 (mod 5). a(n) == {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60).
Terms which are congruent to {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60) and are not members of the sequence: 21, 37, 52, 57, 61, 69, 76, 85, 96, 97, 109, 117, 120, 124, 129, 132, 136, 141, 145, 156, 157, 165, 172, 177, 181, ..., .
Terms which are congruent to {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60), not prime and are not members of the sequence: 21, 52, 57, 69, 76, 85, 96, 117, 120, 124, 129, 132, 136, 141, 145, 156, 165, 172, 177, 184, 192, 201, ..., .
Nonsquare terms: 12, 24, 40, 45, 60, 72, 84, 105, 112, 160, 180, 189, 216, 220, 240, 264, 280, 297, 300, ..., .
The lesser of twin terms: 24, 360, 624, 840, 960, 1104, 1224, 2184, 2400, 2736, ..., .
Lesser term of a gap of n or 0 if impossible: 24, 0, 1, 12, 4, 0, 105, 16, 72, 0, 25, ..., . (End)
Number of terms less than or equal to 10^n: 1, 3, 17, 84, 423, 2123, 10603, 52144, 253257, ..., . - Robert G. Wilson v, Oct 30 2010

			n=1 allows a solution (x,y,z)=(1,1,1), and is in the sequence.
n=4 allows solutions (x,y,z)=(1,2,2) and (2,1,2) and is in the sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..10603 . [From _Robert G. Wilson v_, Oct 25 2010]

fQ[n_] := Block[{c = 0, cong = {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57}, dvs = Divisors@ n, dvt, j = 1, k, lmt1, lmt2}, If[ MemberQ[ cong, Mod[n, 60]], lmtj = Length@ dvs + 1; While[j < lmtj, dvt = Divisors[ n/dvs[[j]]]; k = 1; lmtk = Length@ dvt + 1; While[k < lmtk, If[ dvs[[j]] + dvt[[k]] == n/(dvs[[j]]*dvt[[k]]) + 1, c++ ]; k++ ]; j++ ]]; c > 0]; Select[ Range@ 584, fQ] (* Robert G. Wilson v, Oct 25 2010 *)

is_A171920(n)={ my(L=sqrt(n),yz); fordiv(n,x, x>L & return; fordiv(yz=n/x,y, y>x & break; y*(x+y-1)==yz & return(1)))} \\ M. F. Hasler, Nov 07 2010

More terms from Robert G. Wilson v, Oct 25 2010

A171919 Number of solutions to n=xyz, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2

Offset: 1

Georgi Guninski, Oct 23 2010

nonn

Record values start a(1)=1, a(4)=2, a(112)=3, a(144)=6, a(23400)=8, a(28224)=10.
If n is a perfect square, a(n)>0.
Larger record indices are listed in A181485, and associated values in A181486. - M. F. Hasler, Oct 23 2010
First occurrence of k: 2, 1, 4, 112, 480, 43120, 144, 218880, 23400, ??, 28224, ??, 373464, ??, 247104, ??, 604800, ??, 83010312, ??, 26812800, ..., . - Robert G. Wilson v, Oct 30 2010
a(388778796252000) = 38.

			For n=4, the a(4)=2 solutions are (x,y,z)=(1,2,2) and (2,1,2).
For n=12, the a(12)=1 solution is (x,y,z)=(2,2,3).

Antti Karttunen, Table of n, a(n) for n = 1..28224

Cf. A181485, A181486.

A := proc(n) local a,dvs,x,y,z,dvsyz; a :=0 ; dvs := numtheory[divisors](n) ; for x in dvs do yz := n/x ; dvsyz := numtheory[divisors](yz) ; for y in dvsyz do z := yz/y ; if x+y-z=1 then a := a+1 ; fi; end do; end do:
return a; end proc: seq(A(n),n=1..100) ; # R. J. Mathar, Oct 23 2010

f[n_] := Block[{c = 0, cong = {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57}, dvs = Divisors@ n, dvt, j = 1, k, lmt1, lmt2}, If[ MemberQ[ cong, Mod[n, 60]], lmtj = Length@ dvs + 1; While[j < lmtj, dvt = Divisors[ n/dvs[[j]]]; k = 1; lmtk = Length@ dvt + 1; While[k < lmtk, If[ dvs[[j]] + dvt[[k]] == n/(dvs[[j]]*dvt[[k]]) + 1, c++ ]; k++ ]; j++ ]]; c]; Array[f, 105] (* Robert G. Wilson v, Oct 24 2010 *)

A171919(n)={ my(c=0,t); fordiv(n, z, fordiv( t=n/z, y, y>z & break; y*(z+1-y)==t & c++) ); c} /* can be improved by restricting to x<=y and counting twice if xM. F. Hasler, Oct 23 2010

Some more values from M. F. Hasler, Oct 23 2010

A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]b[2]+a[2]b[1] ].

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 30, 64, 72, 120, 130, 288, 300, 420, 434, 1024, 1040, 1296, 1314, 2400, 2420, 2860, 2882, 6912, 6936, 7800, 7826, 11760, 11788, 13020, 13050, 32768, 32800, 35360, 35394, 46656, 46692, 49932, 49970, 96000, 96040, 101640, 101682

Offset: 1

Georgi Guninski, Sep 21 2010

Define binary operation "o" on pairs of vectors a,b: a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ], define scalar multiplication "x" on vector p: 2n x p = (n x p) o (n x p) (2n+1) x p = ((n x p) o (n x p)) o p 1 x p = p. Sequence is: a(n) = (n x p)[2] for p=[1,1] (the first component is n). Other sequences with o associative?

			Set p=[1,1], a(2)=o(p,p) [2] = [1+1,1*1+1*1] [2]=[2,2] [2]=2; a(3)=o(o(p,p),p) [2]=o([2,2],[1,1]) [2] =[2+1,2*1+1*2] [2] = [3,4] [2] = 4 (note that computation is fast as in fast exponentiation because of the definition of x).

Maximilian Hasler, structure & sequences defined by "exotic multiplication", SeqFan Dec 2009. See the whole thread.

\\ code by M. F. Hasler |vector(20,i,a(i)[2])|
addi(x,y)=[x[1]+y[1],x[1]*y[2]+x[2]*y[1]];
a(n,poi=[1,1])=if(n<=1,poi,if(n%2,n==1&return(poi);addi(a(n-1,poi),poi),poi=a(n\2,poi);addi(poi,poi)))

A180435 a(n) = a(n-1)*2^n+n, a(0)=1.

Original entry on oeis.org

1, 3, 14, 115, 1844, 59013, 3776838, 483435271, 123759429384, 63364827844617, 64885583712887818, 132885675443994251275, 544299726618600453222412, 4458903360459574912797999117, 73054672657769675371282417532942

Offset: 0

Georgi Guninski, Sep 05 2010

nonn

Cf. A010842.

nxt[{n_,a_}]:={n+1,a*2^(n+1)+n+1}; Transpose[NestList[nxt,{0,1},20]] [[2]] (* Harvey P. Dale, Apr 05 2015 *)

PARI
```
a(n)=if(n<=0,1, a(n-1)*2^n+n )
```

a(n+1) = (2^(n + 1) + 1)*a(n) - 2^n*a(n - 1) + 1.
a(n+1) = ((a(n - 2) + 4*a(n - 1) + 4)*a(n) - 2*a(n - 1)^2 - 4*a(n)^2 + a(n - 2) - 4*a(n - 1))/(a(n - 2) - 2*a(n - 1)).
a(n) = 2^(n*(n+1)/2) + sum_{k=1..n} 2^( (n+k+1)*(n-k)/2 ) * k. - Max Alekseyev, Sep 05 2010

Minor edits by N. J. A. Sloane, Sep 05 2010

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User: Georgi Guninski

A182012 Number of graphs on 2n unlabeled nodes all having odd degree.

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A171920 Numbers n with at least one solution to n=x*y*z, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

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A171919 Number of solutions to n=x*y*z, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

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A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ].

Views

Author

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Comments

Examples

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Programs

PARI

A180435 a(n) = a(n-1)*2^n+n, a(0)=1.

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Programs

Mathematica

PARI

Formula

Extensions

A171920 Numbers n with at least one solution to n=xyz, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

A171919 Number of solutions to n=xyz, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]b[2]+a[2]b[1] ].