A010017 a(0) = 1, a(n) = 27*n^2 + 2 for n>0.

1, 29, 110, 245, 434, 677, 974, 1325, 1730, 2189, 2702, 3269, 3890, 4565, 5294, 6077, 6914, 7805, 8750, 9749, 10802, 11909, 13070, 14285, 15554, 16877, 18254, 19685, 21170, 22709, 24302, 25949, 27650, 29405, 31214, 33077, 34994, 36965, 38990, 41069, 43202

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Bruno Berselli, Feb 06 2012: (Start)

First trisection of A005918.

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=5, s=1.

After 1, all terms are in A000408. (End)

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A206399.

Programs

Magma

[1] cat [27*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Mathematica

Join[{1}, 27 Range[40]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
RecurrenceTable[{a[1]==29, a[2]==110, a[3]==245, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3]}, a, {n, 1,30}] (* G. C. Greubel, Aug 02 2015 *)

PARI

first(m)=my(v=vector(m));for(i=1,m,v[i]=27*(i)^2+2);concat([1],v); /* Anders Hellström, Aug 02 2015 */

Formula

G.f.: (1+x)*(1+25*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012

E.g.f.: (x*(x+1)*27+2)*e^x-1. - Gopinath A. R., Feb 14 2012

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), n>=4, a(1)=29, a(2)=110, a(3)=245. - G. C. Greubel, Aug 02 2015

Sum_{n>=0} 1/a(n) = 3/4+sqrt(6)/36*Pi*coth(Pi*sqrt(6)/9) = 1.0581468172342... - R. J. Mathar, May 07 2024