A158563 -id:A158563 - 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

A158575 a(n) = 32*n^2 + 1.

Original entry on oeis.org

1, 33, 129, 289, 513, 801, 1153, 1569, 2049, 2593, 3201, 3873, 4609, 5409, 6273, 7201, 8193, 9249, 10369, 11553, 12801, 14113, 15489, 16929, 18433, 20001, 21633, 23329, 25089, 26913, 28801, 30753, 32769, 34849, 36993, 39201, 41473, 43809, 46209, 48673, 51201, 53793

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (32*n^2 + 1)^2 - (256*n^2 + 16)*(2*n)^2 = 1 can be written as a(n)^2-A158574(n)*A005843(n)^2 = 1. - Comment rewritten by R. J. Mathar, Oct 16 2009

Sequence found by reading the line segment from 1 to 33 together with the line from 33, in the direction 33, 129, ..., in the square spiral whose vertices are the generalized 18-gonal numbers A274979. - Omar E. Pol, Apr 21 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..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. A000217, A005843, A010021, A081585, A158563, A158574, A244082.

Cf. A274979 (generalized 18-gonal numbers).

I:=[1, 33, 129]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

LinearRecurrence[{3, -3, 1}, {1, 33, 129}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
32*Range[0,40]^2+1 (* Harvey P. Dale, Jul 20 2021 *)

for(n=0, 50, print1(32*n^2+1", ")); \\ Vincenzo Librandi, Feb 15 2012

G.f.: (1+30*x+33*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
For n > 0 a(n) = sqrt(8*(A000217(4*n-1)^2 + A000217(4*n)^2) + 1). - J. M. Bergot, Sep 03 2015
a(n) = A244082(n) + 1. - Omar E. Pol, Apr 21 2021
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + coth(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 32*x + 32*x^2).
a(n) = A081585(2*n). (End)

a(0) added by R. J. Mathar, Oct 16 2009

A158562 a(n) = 256*n^2 - 16.

Original entry on oeis.org

240, 1008, 2288, 4080, 6384, 9200, 12528, 16368, 20720, 25584, 30960, 36848, 43248, 50160, 57584, 65520, 73968, 82928, 92400, 102384, 112880, 123888, 135408, 147440, 159984, 173040, 186608, 200688, 215280, 230384, 246000, 262128, 278768, 295920, 313584, 331760

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as A158563(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by R. J. Mathar, Oct 16 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A141759, A158563.

I:=[240,1008,2288]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

16(16Range[40]^2-1) (* or *) LinearRecurrence[{3,-3,1},{240,1008,2288},40] (* Harvey P. Dale, Sep 13 2011 *)

for(n=1, 50, print1(256*n^2-16", ")); \\ Vincenzo Librandi, Feb 15 2012

G.f.: 16*x*(-15 - 18*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=1} 1/a(n) = (4 - Pi)/128.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 4)/128. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 16*(exp(x)*(16*x^2 + 16*x - 1) + 1).
a(n) = 16*A141759(n-1). (End)

A158683 a(n) = 1024*n^2 - 32.

Original entry on oeis.org

992, 4064, 9184, 16352, 25568, 36832, 50144, 65504, 82912, 102368, 123872, 147424, 173024, 200672, 230368, 262112, 295904, 331744, 369632, 409568, 451552, 495584, 541664, 589792, 639968, 692192, 746464, 802784, 861152, 921568, 984032, 1048544, 1115104, 1183712

Offset: 1

Vincenzo Librandi, Mar 24 2009

The identity (64*n^2 - 1)^2 - (1024*n^2 - 32)*(2*n)^2 = 1 can be written as A158684(n)^2 - a(n)*A005843(n)^2 = 1.

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, A158563, A158684.

I:=[992, 4064, 9184]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 19 2012

A158683:=n->1024*n^2-32: seq(A158683(n), n=1..50); # Wesley Ivan Hurt, Nov 20 2014

LinearRecurrence[{3, -3, 1}, {992, 4064, 9184}, 50] (* Vincenzo Librandi, Feb 19 2012 *)

for(n=1, 40, print1(1024*n^2 - 32", ")); \\ Vincenzo Librandi, Feb 19 2012

G.f.: 32*x*(-31-34*x+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>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/64.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/64. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 32*(exp(x)*(32*x^2 + 32*x - 1) + 1).
a(n) = 32*A158563(n). (End)

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

cp's OEIS Frontend

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

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A158575 a(n) = 32*n^2 + 1.

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A158562 a(n) = 256*n^2 - 16.

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A158683 a(n) = 1024*n^2 - 32.

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