A108211 -id:A108211 - cp's OEIS frontend

A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

1, 34, 130, 290, 514, 802, 1154, 1570, 2050, 2594, 3202, 3874, 4610, 5410, 6274, 7202, 8194, 9250, 10370, 11554, 12802, 14114, 15490, 16930, 18434, 20002, 21634, 23330, 25090, 26914, 28802, 30754, 32770, 34850, 36994, 39202, 41474, 43810, 46210, 48674, 51202

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Apr 21 2021: (Start)
Sequence found by reading the line segment from 1 to 34 together with the line from 34, in the direction 34, 130, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
The spiral begins as follows:
       46_   _   _   _   _   _   _   _   _   _18
        |                                                       |
        |                           0                           |
        |                           |  _   _   _   _  |
        |                           1                           15
        |
       51
(End)

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

Cf. A206399, A158563, A158575, A244082.

Cf. A274979 (generalized 18-gonal numbers).

Join[{1}, 32 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
CoefficientList[Series[(1 + x) (1 + 30 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2014 *)

G.f.: (1+x)*(1+30*x+x^2)/(1-x)^3. [Bruno Berselli, Feb 07 2012]
a(n) = A005893(4n) = A008527(2n); a(n+1) = A108100(2n+2). [Bruno Berselli, Feb 07 2012]
E.g.f.: (x*(x+1)*32+2)*e^x-1. - Gopinath A. R., Feb 14 2012
a(n) = (4n+1)^2+(4n-1)^2 for n>0. [Bruno Berselli, Jun 24 2014]
a(n) = A244082(n) + 2, n >= 1. - Omar E. Pol, Apr 21 2021
Sum_{n>=0} 1/a(n) = 3/4 + Pi/16*coth(Pi/4) = 1.04940725316131.. - R. J. Mathar, May 07 2024
a(n) = 2*A108211(n). - R. J. Mathar, May 07 2024
a(n) = A195315(n)+A195315(n+1). - R. J. Mathar, May 07 2024

A158686 a(n) = 64*n^2 + 1.

Original entry on oeis.org

1, 65, 257, 577, 1025, 1601, 2305, 3137, 4097, 5185, 6401, 7745, 9217, 10817, 12545, 14401, 16385, 18497, 20737, 23105, 25601, 28225, 30977, 33857, 36865, 40001, 43265, 46657, 50177, 53825, 57601, 61505, 65537, 69697, 73985, 78401, 82945, 87617, 92417, 97345, 102401

Offset: 0

Vincenzo Librandi, Mar 24 2009

The identity (64*n^2 + 1)^2 - (1024*n^2 + 32)*(2*n)^2 = 1 can be written as a(n)^2 - A158685(n)*(A005843(n))^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A108211, A158685.

[64*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013

64 Range[0, 40]^2 + 1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 65, 257}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[- (1 + 62 x + 65 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

a(n)=64*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -(1+62*x+65*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 21 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/8)*Pi/8 + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/8)*Pi/8 + 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(1 + 64*x + 64*x^2).
a(n) = A108211(2*n) for n > 0. (End)

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

A158488 a(n) = 64*n^2 + 8.

Original entry on oeis.org

72, 264, 584, 1032, 1608, 2312, 3144, 4104, 5192, 6408, 7752, 9224, 10824, 12552, 14408, 16392, 18504, 20744, 23112, 25608, 28232, 30984, 33864, 36872, 40008, 43272, 46664, 50184, 53832, 57608, 61512, 65544, 69704, 73992, 78408, 82952, 87624, 92424, 97352, 102408

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (16*n^2 + 1)^2 - (64*n^2 + 8)*(2*n)^2 = 1 can be written as A108211(n)^2 - a(n)*A005843(n)^2 = 1. - rewritten by Bruno Berselli, Nov 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A081585, A108211.

I:=[72,264,584]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012

A158488:=n->64*n^2+8: seq(A158488(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2017

64Range[40]^2+8 (* or *) LinearRecurrence[{3,-3,1},{72,264,584},40] (* Harvey P. Dale, Nov 16 2011 *)

for(n=1, 40, print1(64*n^2 + 8", ")); \\ Vincenzo Librandi, Feb 08 2012

a(1)=72, a(2)=264, a(3)=584, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Nov 16 2011
G.f: x*(72 + 48*x + 8*x^2)/(1-x)^3. - Vincenzo Librandi, Feb 08 2012
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)) - 1)/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)))/16. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 8*(exp(x)*(8*x^2 + 8*x + 1) - 1).
a(n) = 8*A081585(n). (End)

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A010021 a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

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A158686 a(n) = 64*n^2 + 1.

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A158488 a(n) = 64*n^2 + 8.

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