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A010000 a(0) = 1, a(n) = n^2 + 2 for n>0. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

G.f.: (1 + x^3) / (1 - x)^3. a(n) = a(-n) for all n in Z. - Michael Somos, Aug 11 2009

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

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

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

Join[{1}, Range[50]^2 + 2] (* Bruno Berselli, Feb 08 2012 *)

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)^(n-j)*binomial(n, j)*(j+1/n)^(n+1), j, 0, n)/(n+1)!; \\ Vladimir Kruchinin, Jun 03 2013

CROSSREFS

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

Cf. A206399.

Sequence in context: A140126 A140235 A224214 * A183199 A172046 A014125

Adjacent sequences:  A009997 A009998 A009999 * A010001 A010002 A010003

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 25 19:28 EDT 2017. Contains 287059 sequences.