A140145 -id:A140145 - cp's OEIS frontend

A135276 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^1 if n is even.

0, 1, 3, 4, 8, 9, 15, 16, 24, 25, 35, 36, 48, 49, 63, 64, 80, 81, 99, 100, 120, 121, 143, 144, 168, 169, 195, 196, 224, 225, 255, 256, 288, 289, 323, 324, 360, 361, 399, 400, 440, 441, 483, 484, 528, 529, 575, 576, 624, 625, 675, 676, 728, 729, 783, 784, 840, 841, 899, 900, 960, 961

Offset: 0

Artur Jasinski, May 12 2008, corrected May 17 2008

Index to family of sequences of the form a(n) = a(n-1) + n^r if n odd, a(n) = a(n-1)+ n^s if n is even, for n > 1 and a(1)=1:
         s=0,     s=1,      s=2,     s=3,     s=4,     s=5
  r=0, A000027, this seq, A135301, A135332, A140142, A140143;
  r=1, A140144, A000217,  A140113, A140145, A140146, A140147;
  r=2, A140148, A136047,  A000330, A140149, A140150, A140151;
  r=3, A140152, A140153,  A140154, A000537, A140155, A140156;
  r=4, A140157, A140158,  A140159, A140160, A000538, A140161;
  r=5, A140162, A140163,  A135095, A135099, A135214, A000539.
Equals triangle A070909 * [1,2,3,...]. - Gary W. Adamson, May 16 2010
Right edge of the triangle in A199332: a(n) = A199332(n,n), for n > 0. - Reinhard Zumkeller, Nov 23 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A004526, A136047, A140113.

Cf. A070909. - Gary W. Adamson, May 16 2010

[(2*n^2+6*n+1+(2*n-1)*(-1)^n)/8 : n in [0..100]]; // Wesley Ivan Hurt, Mar 22 2016

A135276:=n->( 2*n^2 + 6*n + 1 + (2*n-1)*(-1)^n )/8: seq(A135276(n), n=0..100); # Wesley Ivan Hurt, Mar 22 2016

a = {}; r = 0; s = 1; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 3, 4, 8}, 50] (* G. C. Greubel, Oct 08 2016 *)

A135276(n)=if(n%2,((n+1)/2)^2,(n/2+1)^2-1) \\ M. F. Hasler, May 17 2008

my(x='x+O('x^200)); concat(0, Vec(x*(1+2*x-x^2)/((1+x)^2*(1-x)^3))) \\ Altug Alkan, Mar 23 2016

a(n) = (n/2 + 1)^2 - 1 if n is even, ((n+1)/2)^2 if n is odd. - M. F. Hasler, May 17 2008
From R. J. Mathar, Feb 22 2009: (Start)
G.f.: x*(1+2*x-x^2)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
a(n) = (2*n^2 + 6*n + 1 + (2*n-1)*(-1)^n)/8. - Luce ETIENNE, Jul 08 2014
a(n) = (floor(n/2)+1)^2 + (n mod 2) - 1. - Wesley Ivan Hurt, Mar 22 2016
a(n) = A004526((n+1)^2) - A004526(n+1)^2. - Bruno Berselli, Oct 21 2016
Sum_{n>=1} 1/a(n) = 3/4 + Pi^2/6. - Amiram Eldar, Sep 08 2022

Offset corrected by R. J. Mathar, Feb 22 2009

Edited by Michel Marcus, Apr 07 2023

A138237 Number of unlabeled graphs with at least one cycle in which every connected component has at most one cycle.

Original entry on oeis.org

1, 3, 9, 26, 71, 197, 543, 1507, 4186, 11722, 32883, 92724, 262179, 743792, 2115019, 6028779, 17217093, 49258009, 141142096, 404997704, 1163569094, 3346830818, 9636723582, 27774427243, 80121104084, 231317022483, 668346261557

Offset: 3

Washington Bomfim, May 17 2008

nonn

			a(9)=543 since we have several cases, with one unicyclic graph, or two, or three. Namely,
-One triangle and a forest of order 6, or 20 graphs.
-One unicyclic graph with 4 nodes and a forest of order 5, or 20 graphs.
-One unicyclic graph with 5 nodes and a forest of order 4, or 30 graphs.
-One unicyclic graph with 6 nodes and a forest of order 3, or 39 graphs.
-One unicyclic graph of 7 nodes and a forest of order 2, or 66 graphs.
-One unicyclic graph of 8 nodes and an isolated vertex, or 89 graphs.
-One unicyclic graph of 9 nodes, or 240 graphs.
-Two triangles and a forest of order 3, or 3 graphs.
-One triangle plus one unicyclic graph of 4 nodes plus a forest of order 2, or 4 graphs.
-One triangle plus one unicyclic graph of 5 nodes plus an isolated vertex, or 5 graphs.
-One triangle plus one unicyclic graph of 6 nodes, or 13 graphs.
-Two unicyclic graphs of 4 nodes and an isolated vertex, or C(2+2-1,2)=3 graphs.
-One unicyclic graph of 5 nodes and another of 4 nodes, or 10 graphs.
-Three triangles, or 1 graph.
Total = 543.

Wikipedia, Pseudoforest.

Cf. A001429, A005195, A140145.

a(n) = A134964(n) - A005195(n).

cp's OEIS Frontend

A135276 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^1 if n is even.

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