A158575 -id:A158575 - 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

A158563 a(n) = 32*n^2 - 1.

Original entry on oeis.org

31, 127, 287, 511, 799, 1151, 1567, 2047, 2591, 3199, 3871, 4607, 5407, 6271, 7199, 8191, 9247, 10367, 11551, 12799, 14111, 15487, 16927, 18431, 19999, 21631, 23327, 25087, 26911, 28799, 30751, 32767, 34847, 36991, 39199, 41471, 43807, 46207, 48671, 51199, 53791

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 a(n)^2 - A158562(n)*A005843(n)^2 = 1. [comment rewritten by R. J. Mathar, Oct 16 2009]
From Omar E. Pol, Apr 21 2021: (Start)
Sequence found by reading the line from 31, in the direction 31, 127, ..., 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)

Vincenzo Librandi, Table of n, a(n) for n = 1..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, A010021, A158562, A158575, A244082.

Cf. A274979 (generalized 18-gonal numbers).

[32*n^2-1: n in [1..40]]; // Vincenzo Librandi, Sep 11 2013

32 Range[40]^2 - 1  (* Harvey P. Dale, Mar 04 2011 *)
CoefficientList[Series[(- 31 - 34 x + x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

a(n)=32*n^2-1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(-31-34*x+x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A244082(n) - 1. - Omar E. Pol, Apr 21 2021
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/2. (End)
E.g.f.: 1 + exp(x)*(32*x^2 + 32*x - 1). - Elmo R. Oliveira, Jan 25 2025

A158574 a(n) = 256*n^2 + 16.

Original entry on oeis.org

16, 272, 1040, 2320, 4112, 6416, 9232, 12560, 16400, 20752, 25616, 30992, 36880, 43280, 50192, 57616, 65552, 74000, 82960, 92432, 102416, 112912, 123920, 135440, 147472, 160016, 173072, 186640, 200720, 215312, 230416, 246032, 262160, 278800, 295952, 313616, 331792

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 A158575(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158575.

I:=[16, 272, 1040]; [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}, {16, 272, 1040}, 50] (* Vincenzo Librandi, Feb 15 2012 *)

for(n=0, 50, print1(256*n+16", ")); \\ Vincenzo Librandi, Feb 15 2012

G.f.: 16*(1 + 14*x + 17*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/4)*Pi/4 + 1)/32.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/4)*Pi/4 + 1)/32. (End)

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

A158685 a(n) = 32(32n^2 + 1).

Original entry on oeis.org

32, 1056, 4128, 9248, 16416, 25632, 36896, 50208, 65568, 82976, 102432, 123936, 147488, 173088, 200736, 230432, 262176, 295968, 331808, 369696, 409632, 451616, 495648, 541728, 589856, 640032, 692256, 746528, 802848, 861216, 921632, 984096, 1048608, 1115168, 1183776

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 A158686(n)^2 - a(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, A158575, A158686.

[32*(32*n^2+1): n in [0..40]]; // Vincenzo Librandi, Sep 11 2013

CoefficientList[Series[ - 32 (1 + 30 x + 33 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

a(n)=32*(32*n^2+1) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -32*(1+30*x+33*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/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 32*exp(x)*(1 + 32*x + 32*x^2).
a(n) = 32*A158575(n). (End)

Comment rewritten, a(0) added 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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A158563 a(n) = 32*n^2 - 1.

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A158574 a(n) = 256*n^2 + 16.

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A158685 a(n) = 32*(32*n^2 + 1).

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A158685 a(n) = 32(32n^2 + 1).