A010000 a(0) = 1, a(n) = n^2 + 2 for n > 0.

1, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Least k such that A070864(k) = 2n-1. - Robert G. Wilson v and Benoit Cloitre, May 20 2002

With an offset of 3, beginning with 6 (deleting first two terms) n*(n+a(n)) + 1 is a cube = (n+1)^3: 1*(1+6) + 1 = 8, 2*(2+11) + 1 = 27, etc. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

For n >= 2, a(n) is the maximum element in the continued fraction for Sum_{k>=1} 1/n^(2^k) (for n=2 see A006464). - Benoit Cloitre, Jun 12 2007

Equals binomial transform of [1, 2, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008

Minimum Wiener index of 3-degenerate graphs with n+2 vertices. A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are maximal 3-degenerate graphs with diameter at most 2. - Allan Bickle, Oct 14 2022

a(n-1) is the number of unit triangles enclosed by the triangular spiral drawn on a isometric grid of which the n-th side has length n. The picture in the link shows how the spiral is constructed. - Bob Andriesse, Feb 14 2023

Examples

			G.f. = 1 + 3*x + 6*x^2 + 11*x^3 + 18*x^4 + 27*x^5 + 38*x^6 + 51*x^7 + 66*x^8 + ...
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Bob Andriesse, Graphical representation of the first 21 segments of the spiral with grid.
• Allan Bickle, Wiener indices of maximal k-degenerate graphs, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
• Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A070864. Apart from initial terms, same as A059100.

Cf. A206399.

Cf. A002061 (minimum Wiener index of 2-degenerate graphs).

Programs

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = 2 + a[n - a[n - 1]]; b = Table[0, {100}]; Do[c = (a[n] + 1)/2; If[c < 101 && b[[c]] == 0, b[[c]] = n], {n, 1, 10^4}]; b
Join[{1}, Range[50]^2 + 2] (* Bruno Berselli, Feb 08 2012 *)
a[ n_] := n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 05 2015 *)

Maxima

a(n):=if n=0 then 1 else 2*n*sum((-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1),j,0,n)/(n+1)!; /* Vladimir Kruchinin, Jun 03 2013 */

PARI

{a(n) = n^2 + 2 - (n==0)}; /* Michael Somos, Aug 11 2009 */

Formula

a(n) = A000217(n-2) + A000217(n+1) for n > 0. - Jon Perry, Jul 23 2003

Euler transform of length 6 sequence [ 3, 0, 1, 0, 0, -1]. - Michael Somos, Aug 11 2009

From Michael Somos, Aug 11 2009: (Start)

G.f.: (1 + x^3) / (1 - x)^3.

a(n) = a(-n) for all n in Z. (End)

E.g.f.: (x*(x+1) + 2)*e^x - 1. - Gopinath A. R., Feb 14 2012

a(n) = (2*n/(n+1)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 03 2013

Sum_{n>=0} 1/a(n) = 3/4+1/4*sqrt(2)*Pi*coth(Pi*sqrt 2)= 1.8610281... - R. J. Mathar, May 07 2024