A058788 -id:A058788 - cp's OEIS frontend

A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25, 24, 25, 26, 25, 26, 27, 26, 27, 28

Offset: 0

N. J. A. Sloane, Mar 15 1996

Molien series of 2-dimensional representation of cyclic group of order 3 over GF(2).
One step back, two steps forward.
The crossing number of the graph C(n, {1,3}), n >= 8, is [n/3] + n mod 3, which gives this sequence starting at the first 4. [Yang Yuansheng et al.]
A Chebyshev transform of A078008. The g.f. is the image of (1-x)/(1-x-2*x^2) (g.f. of A078008) under the Chebyshev transform A(x)-> (1/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 15 2004
A047878 is an essentially identical sequence. - Anton Chupin, Oct 24 2009
Rhyme scheme of Dante Alighieri's "Divine Comedy." - David Gaita, Feb 11 2011
A194960 results from deleting the first four terms of A008611. Note that deleting the first term or first four terms of A008611 leaves a concatenation of segments (n, n+1, n+2); for related concatenations, see
  A008619, (n,n+1) after deletion of first term;
  A053737, (n,n+1,n+2,n+3) beginning with n=0;
  A053824, (n to n+4) beginning with n=0. - Clark Kimberling, Sep 07 2011
It appears that a(n) is the number of roots of x^(n+1) + x + 1 inside the unit circle. - Michel Lagneau, Nov 02 2012
Also apparently for n >= 2: a(n) is the largest remainder r that results from dividing n+2 by 1..n+2 more than once, i.e., a(n) = max(i, A072528(n+2,i)>1). - Ralf Stephan, Oct 21 2013
Number of n-element subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}. - Alois P. Heinz, Feb 06 2017
It appears that a(n) is the number of roots of the Fibonacci polynomial F(n+2,x) strictly inside the unit circle of the complex plane. - Michel Lagneau, Apr 07 2017
For the proof of the preceding conjecture see my comments under A008615 and A049310. Chebyshev S(n,x) = i^n*F(n+1,-i*x), with i = sqrt(-1). - Wolfdieter Lang, May 06 2017
The sequence is the interleaving of three sequences: the positive integers (A000027), the nonnegative integers (A001477), and the positive integers, in that order. - Guenther Schrack, Nov 07 2020
a(n) is the number of multiples of 3 between n and 2n. - Christian Barrientos, Dec 20 2021
a(n) is the least number of football games a team has to play to be able to get n-1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points. - Sigurd Kittilsen, Dec 01 2022

			G.f. = 1 + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 103.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Frederik Glitzner and David Manlove, Perspectives on Unsolvability in Roommates Markets, arXiv:2505.06717 [cs.GT], 2025. See p. 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 447.
Gerard P. Michon, Counting Polyhedra.
Yang Yuansheng et al., The crossing number of C(n; {1,3}), Discr. Math. 289 (2004), 107-118.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for Molien series.

Cf. A000027, A001477, A008619, A047878, A049310, A049347, A053737, A053824, A058207, A058788, A072528, A078008, A194960, A257075, A330396.

a008611 n = n' + mod r 2 where (n', r) = divMod (n + 1) 3
a008611_list = f [1,0,1] where f xs = xs ++ f (map (+ 1) xs)
-- Reinhard Zumkeller, Nov 25 2013

[(n-1)-2*Floor((n-1)/3): n in [0..90]]; // Vincenzo Librandi, Aug 21 2011

with(numtheory): for n from 1 to 70 do:it:=0:
y:=[fsolve(x^n+x+1, x, complex)] : for m from 1 to nops(y) do : if abs(y[m])< 1 then it:=it+1:else fi:od: printf(`%d, `,it):od:
A008611:=n->(n-1)-2*floor((n-1)/3); seq(A008611(n), n=0..50); # Wesley Ivan Hurt, May 18 2014

With[{nn=30},Riffle[Riffle[Range[nn],Range[0,nn-1]],Range[nn],3]] (* or *) RecurrenceTable[{a[0]==a[2]==1,a[1]==0,a[n]==a[n-3]+1},a,{n,90}] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
a[ n_] := Quotient[n - 1, 3] + Mod[n + 2, 3]; (* Michael Somos, Jan 23 2014 *)

{a(n) = (n-1) \ 3 + (n+2) % 3}; /* Michael Somos, Jan 23 2014 */

a(n) = a(n-3) + 1.
a(n) = (n-1) - 2*floor((n-1)/3).
G.f.: (1 + x^2 + x^4)/(1 - x^3)^2.
After the initial term, has form {n, n+1, n+2} for n=0, 1, 2, ...
From Paul Barry, Mar 18 2004: (Start)
a(n) = Sum_{k=0..n} (-1)^floor(2*(k-2)/3);
a(n) = 4*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3. (End)
From Paul Barry, Oct 15 2004: (Start)
G.f.: (1 - x + x^2)/((1 + x + x^2)*(x-1)^2);
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A078008(n-2k)*(-1)^k. (End)
a(n) = -a(-2-n) for all n in Z.
Euler transform of length 6 sequence [0, 1, 2, 0, 0, -1]. - Michael Somos, Jan 23 2014
a(n) = ((n-1) mod 3) + floor((n-1)/3). - Wesley Ivan Hurt, May 18 2014
PSUM transform of A257075. - Michael Somos, Apr 15 2015
a(n) = A194960(n-3), n >= 0, with extended A194960. See the a(n) formula two lines above. - Wolfdieter Lang, May 06 2017
From Guenther Schrack, Nov 07 2020: (Start)
a(n) = (3*n + 3 + 2*(w^(2*n)*(1 - w) + w^n*(2 + w)))/9, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
a(n) = (n + 1 + 2*A049347(n))/3;
a(n) = (2*n - A330396(n-1))/3. (End)
E.g.f.: (3*exp(x)*(1 + x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, May 06 2022
Sum_{n>=2} (-1)^n/a(n) = 3*log(2) - 1. - Amiram Eldar, Sep 10 2023

A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 8, 11, 8, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50

Offset: 1

Brendan McKay

			Triangle begins
  0;
  0,0;
  0,0,0,0;
  0,0,0,1,0,0,0;
  0,0,0,0,0,1,1,0,0,0;
  0,0,0,0,0,0,1,2,2,...;
  ...
From _Hugo Pfoertner_, Nov 24 2020: (Start)
Transposed table:
.
                              Nodes                        Sums
       4  5  6   7   8    9    10     11     12    13  14 |A002840
  Edges-+--+--+---+---+----+-----+------+------+-----+---+|-------
   6 | 1  .  .   .   .    .     .      .      .     .   . |      1
   7 | .  .  .   .   .    .     .      .      .     .   . |      0
   8 | .  1  .   .   .    .     .      .      .     .   . |      1
   9 | .  1  1   .   .    .     .      .      .     .   . |      2
  10 | .  .  2   .   .    .     .      .      .     .   . |      2
  11 | .  .  2   2   .    .     .      .      .     .   . |      4
  12 | .  .  2   8   2    .     .      .      .     .   . |     12
  13 | .  .  .  11  11    .     .      .      .     .   . |     22
  14 | .  .  .   8  42    8     .      .      .     .   . |     58
  15 | .  .  .   5  74   74     5      .      .     .   . |    158
  16 | .  .  .   .  76  296    76      .      .     .   . |    448
  17 | .  .  .   .  38  633   633     38      .     .   . |   1342
  18 | .  .  .   .  14  768  2635    768     14     .   . |   4199
  19 | .  .  .   .   .  538  6134   6134    558     .   . |  13384
  20 | .  .  .   .   .  219  8822  25626   8822   219   . |  43708
  21 | .  .  .   .   .   50  7916  64439  64439  7916  50 | 144810
  .. | .  .  .   .   .    .    ..     ..     ..    ..  .. |     ..
     ---+--+--+---+---+----+-----+------+-------+----+---+
  Sums 1  2  7  34 257 2606 32300 440564 6384634 .. A000944
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..469; rows 1..14 flattened

Cf. A000944, A002840, A021103, A003094, A049334.
A049337, A058787, A212438 are all versions of the same triangle.
Cf. A058788.

Missing zeros inserted by Sean A. Irvine, Jul 29 2021

A058787 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n faces and k vertices, where (n/2+2) <= k <= (2n+8).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 8, 11, 8, 5, 2, 11, 42, 74, 76, 38, 14, 8, 74, 296, 633, 768, 558, 219, 50, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249, 14, 558, 8822, 64439, 268394, 709302

Offset: 4

Gerard P. Michon, Nov 29 2000

Rows are of lengths 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, ... floor(3n/2)-5. See A001651 (this is the sequence of integers not divisible by 3).

			There are 38 polyhedra with 9 faces and 11 vertices, or with 11 faces and 9 vertices.

G. P. Michon, Counting Polyhedra

Cf. A000109, A002856, A000944, A002840, A058786, A058788, A001651.

A049337, A058787, A212438 are all versions of the same triangle.

A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

Original entry on oeis.org

1, 2, 8, 38, 219, 1404, 9714, 70454, 527235, 4037671, 31477887, 249026400, 1994599707, 16147744792, 131959532817, 1087376999834, 9027039627035, 75441790558926, 634311771606750, 5362639252793358, 45565021714371644, 388937603694422120, 3333984869758146814

Offset: 5

Gerard P. Michon, Nov 29 2000

			a(5)=1 because the square pyramid is the only pentahedron with 5=2*5-5 vertices (or 8=3*5-7 edges). Alternatively, a(5)=1 because the square pyramid is the only polyhedron with 5 vertices whose faces are all triangles with only one tetragonal exception.

Andrew Howroyd, Table of n, a(n) for n = 5..500
CombOS - Combinatorial Object Server, generate planar graphs
G. P. Michon, Counting Polyhedra

Column k=4 of A342053.

Cf. A000109, A002856, A000944, A002840, A058787, A058788, A049337.

A342053ColSeq(25,4) \\ See links in A342053. - Andrew Howroyd, Feb 28 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 27 2021

cp's OEIS Frontend

A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

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A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

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A058787 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n faces and k vertices, where (n/2+2) <= k <= (2n+8).

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A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

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A058789 Number of polyhedra with n faces and n+1 vertices (or n vertices and n+1 faces).

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