

A010000


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


17



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
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OFFSET

0,2


COMMENTS

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 3degenerate graphs with n+2 vertices. A maximal 3degenerate graph can be constructed from a 3clique by iteratively adding a new 3leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are maximal 3degenerate graphs with diameter at most 2.  Allan Bickle, Oct 14 2022
a(n1) is the number of unit triangles enclosed by the triangular spiral drawn on a isometric grid of which the nth side has length n. The picture in the link shows how the spiral is constructed.  Bob Andriesse, Feb 14 2023


LINKS



FORMULA

Euler transform of length 6 sequence [ 3, 0, 1, 0, 0, 1].  Michael Somos, Aug 11 2009
G.f.: (1 + x^3) / (1  x)^3.
a(n) = a(n) for all n in Z. (End)
a(n) = (2*n/(n+1)!)*Sum_{j=0..n} (1)^(nj)*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


EXAMPLE

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 + ...


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
a[ n_] := n^2 + 2  Boole[n == 0]; (* Michael Somos, May 05 2015 *)


PROG

(PARI) {a(n) = n^2 + 2  (n==0)}; /* Michael Somos, Aug 11 2009 */
(Maxima)
a(n):=if n=0 then 1 else 2*n*sum((1)^(nj)*binomial(n, j)*(j+1/n)^(n+1), j, 0, n)/(n+1)!; \\ Vladimir Kruchinin, Jun 03 2013


CROSSREFS

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


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



