A008594 -id:A008594 - cp's OEIS frontend

A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

24, 240, 864, 2112, 4200, 7344, 11760, 17664, 25272, 34800, 46464, 60480, 77064, 96432, 118800, 144384, 173400, 206064, 242592, 283200, 328104, 377520, 431664, 490752, 555000, 624624, 699840, 780864, 867912, 961200

Offset: 1

Eric W. Weisstein, Jul 10 2011

Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000326, A000330, A000292, A017233, A008594, A140676, A046092.

a(n) = 12*n^2*(3*n-1).
a(n) = 24*A050509(n).
G.f.: 24*x*(2*x^2+6*x+1)/(x-1)^4. [Colin Barker, Nov 04 2012]
From Bruce J. Nicholson, Sep 18 2019: (Start)
a(n) = 24*n * A000326(n).
a(n) = 4*n^2 * A017233(n).
a(n) = 24*(n^3 + A000292(n-2) + A000330(n-2)).
a(n) = 24*(n^4 - (A008585(n) * A000330(n-1))).
a(n) = 6*A046092(n) + (A008594(n+1) * A140676(n-1)). (End)

A242570 a(n) = 252 * n.

Original entry on oeis.org

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

Offset: 0

Derek Orr, May 17 2014

As lcm(1,2,3,...,9) = 2520, 10*a(n) + k is divisible by each k from 1 through 9.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

Cf. A044102, A135628.

252*Range[0, 49] (* Alonso del Arte, May 17 2014 *)
LinearRecurrence[{2,-1},{0,252},50] (* Harvey P. Dale, Mar 25 2025 *)

PARI
```
for(n=0,50,print(252*n))
```

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 252*x/(x-1)^2.
E.g.f.: 252*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

Original entry on oeis.org

3, 4, 8, 9, 15, 16, 20, 21, 27, 28, 32, 33, 39, 40, 44, 45, 51, 52, 56, 57, 63, 64, 68, 69, 75, 76, 80, 81, 87, 88, 92, 93, 99, 100, 104, 105, 111, 112, 116, 117, 123, 124, 128, 129, 135, 136, 140, 141, 147, 148, 152, 153, 159, 160, 164, 165, 171, 172, 176, 177

Offset: 1

Paolo P. Lava, Apr 19 2017

Numbers with no sum equal to n are listed in A108118, with two sums equal to n are listed in A017593 and with three sums equal to n in A008594.
First difference has period 4: {1,4,1,6}.
Numbers that are congruent to {3, 4, 8, 9} mod 12. - Amiram Eldar, Dec 31 2021

			20 is in the sequence because:
gcd(k,20) = 1 for k = 1, 3, 7, 9, 11, 13, 17, 19: sum is 80.
gcd(k,20) = 2 for k = 2, 6, 14, 18: sum is 40.
gcd(k,20) = 4 for k = 4, 8, 12, 16: sum is 40.
gcd(k,20) = 5 for k = 5, 15: sum is 20.
gcd(k,20) = 10 for k = 10: sum is 10.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008594, A017593, A108118.

P:=proc(q) local a,k,n,t;
for n from 1 to q do a:=array(1..n-1); for k from 1 to n-1 do a[k]:=0; od;
for k from 1 to n-1 do a[gcd(n,k)]:=a[gcd(n,k)]+k; od; t:=0;
for k from 1 to n-1 do if a[k]=n then t:=t+1; fi; od; if t=1 then print(n); fi;
od; end: P(10^6);

Flatten@ Position[#, k_ /; Length@ k == 1] &@ Table[Select[Transpose@ {Values@ #, Keys@ #} &@ Map[Total, PositionIndex@ Map[GCD @@ {n, #} &, Range[n - 1]]], First@ # == n &][[All, -1]], {n, 180}] (* Michael De Vlieger, Apr 28 2017, Version 10 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 8, 9, 15}, 60] (* Amiram Eldar, Dec 31 2021 *)

a(n) = n--; [3, 4, 8, 9][n%4+1] + 12*(n\4) \\ David A. Corneth, Apr 28 2017

is(n) = {my(d=divisors(n), map=vector(d[#d-1]), v=vector(#d-1)); for(i=1,#d-1, map[d[i]]=i); for(i=1,n-1,v[map[gcd(i, n)]]+=i); sum(i=1,#v,v[i]==n)==1} \\ David A. Corneth, Apr 28 2017

is(n) = vecsort(concat([3, 4, 8, 9], [n%12]), ,8)==[3, 4, 8, 9] \\ David A. Corneth, Apr 28 2017

From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x*(3*x^4 + x^3 + 4*x^2 + x + 3)/(x^5 - x^4 - x + 1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(3))*Pi/36. - Amiram Eldar, Dec 31 2021

A338259 Triangle read by rows: T(n,k) is the coefficient of (1+x)^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,n).

Original entry on oeis.org

1, 12, 18, 41, 24, 120, 200, 120, 24, 11, 36, 306, 996, 1446, 984, 303, 42, 21, 48, 576, 2800, 6525, 7848, 4957, 1644, 274, 22, 11, 60, 930, 6020, 19365, 33600, 32487, 17694, 5336, 858, 71, 21, 72, 1368, 11064, 45435, 103200, 134806, 102912, 45567, 11358, 1510, 86, 1

Offset: 1

Henryk A. Witek, Oct 19 2020

The maximum k for which T(n,k) is nonzero, denoted by Cl(n), is usually referred to as the Clar number of O(3,4,n); one has: Cl(1)=3, Cl(2)=6, Cl(3)=8, Cl(4)=10,  Cl(5)=11, and Cl(n)=12 for n>5.
T(n,k) denotes the number of perfect matchings (i.e., Kekulé structures) with k proper sextets for the hexagonal graphene flake O(3,4,n).
ZZ polynomials of hexagonal graphene flakes O(3,4,n) can be computed using ZZDecomposer (see link below), a graphical program to compute ZZ polynomials of benzenoids, or using ZZCalculator (see link below).

			Triangle begins:
   k=0 k=1 k=2   k=3    k=4    k=5    k=6    k=7   k=8   k=9 k=10 k=11 k=12
n=1: 1 12   18     4
n=2: 1 24  120   200    120     24      1
n=3: 1 36  306   996   1446    984    303     42     2
n=4: 1 48  576  2800   6525   7848   4957   1644   274    22    1
n=5: 1 60  930  6020  19365  33600  32487  17694  5336   858   71   2
n=6: 1 72 1368 11064  45435 103200 134806 102912 45567 11358 1510  86  1
   ...
Row n=4 corresponds to the polynomial 1 + 48*(1+x) + 576*(1+x)^2 + 2800*(1+x)^3 + 6525*(1+x)^4 + 7848*(1+x)^5 + 4957*(1+x)^6 + 1644*(1+x)^7 + 274*(1+x)^8 + 22*(1+x)^9 + (1+x)^10.

C.-P. Chou, ZZDecomposer executable.
C.-P. Chou, ZZCalculator source code.
C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH Commun. Math. Comput. Chem. 68 (2012), 3-30.
C.-P. Chou and H. A. Witek, ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 105 for a graphical definition of O(3,4,n)).
H. Zhang and F. Zhang, The Clar covering polynomial of hexagonal systems III, Discrete Math. 212 (2000), 261-269 (proper sextet is defined in Fig.1 and ZZ polynomial in the basis of (1+x)^k monomials is defined by Theorem 2).

Column k=0 is A000012.
Column k=1 is A008594.
Row n=3 is identical to row n=4 of A338217 owing to symmetry of hexagonal graphene flakes.
Row sums give A107915.
Row sums give column k=0 of A338244.

(n,k) -> binomial(n,k)*binomial(12,k)+18*binomial(n+1,k)*binomial(10,k-2)+84*binomial(n+2,k)*binomial(8,k-4)+126*binomial(n+3,k)*binomial(6,k-6)+57*binomial(n+4,k)*binomial(4,k-8)+4*binomial(n+5,k)*binomial(2,k-10) +add(4*binomial(n+1+h,k)*binomial(9,k-3)+24*binomial(n+2+h,k)*binomial(7,k-5)+36*binomial(n+3+h,k)*binomial(5,k-7)+14*binomial(n+4+h,k)*binomial(3,k-9),h = 0 .. 1) +add(add(binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k)*binomial(6-2*s,k-6-2*s),s = 0 .. 2),h = 0 .. 2)

T(n,k) = binomial(n,k)*binomial(12,k) + 18*binomial(n+1,k)*binomial(10,k-2) + 84*binomial(n+2,k)*binomial(8,k-4) + 126*binomial(n+3,k)*binomial(6,k-6) + 57*binomial(n+4,k)*binomial(4,k-8) + 4*binomial(n+5,k)*binomial(2,k-10) + Sum_{h=0..1} (4*binomial(n+1+h,k)*binomial(9,k-3) + 24*binomial(n+2+h,k)*binomial(7,k-5) + 36*binomial(n+3+h,k)*binomial(5,k-7) + 14*binomial(n+4+h,k)*binomial(3,k-9)) + Sum_{s=0..2} Sum_{h=0..2} binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k)*binomial(6-2*s,k-6-2*s) (conjectured, explicitly confirmed for n=1..1000).

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A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

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A242570 a(n) = 252 * n.

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A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

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Author

Keywords

Comments

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Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A338259 Triangle read by rows: T(n,k) is the coefficient of (1+x)^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,n).

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Examples

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Maple

Formula