A108100 -id:A108100 - cp's OEIS frontend

A008527 Coordination sequence for body-centered tetragonal lattice.

1, 10, 34, 74, 130, 202, 290, 394, 514, 650, 802, 970, 1154, 1354, 1570, 1802, 2050, 2314, 2594, 2890, 3202, 3530, 3874, 4234, 4610, 5002, 5410, 5834, 6274, 6730, 7202, 7690, 8194, 8714, 9250, 9802, 10370, 10954, 11554, 12170, 12802, 13450, 14114, 14794, 15490, 16202, 16930, 17674

Offset: 0

N. J. A. Sloane

Also sequence found by reading the segment (1, 10) together with the line from 10, in the direction 10, 34, ..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Apart from leading term, same as A108100.
Cf. A206399.
Cf. A016754 (SE), A054554 (NE), A054569 (SW), A053755 (NW), A033951 (S), A054552 (E), A054556 (N), A054567 (W) (Ulam spiral spokes).
A143839 (SSE) + A143855 (ESE) = A143838 (SSW) + A143856 (ENE) = A143854 (WSW) + A143861 (NNE) = A143859 (WNW) + A143860 (NNW) = even bisection = a(2n) = A010021(n).

Concatenation([1], List([1..40], n-> 2*(1+4*n^2) )); # G. C. Greubel, Nov 09 2019

[1] cat [2*(1 + 4*n^2): n in [1..50]]; // G. C. Greubel, Nov 09 2019

Maple
```
1, seq(8*k^2+2, k=1..50);
```

a[0]:= 1; a[n_]:= 8n^2 +2; Table[a[n], {n,0,50}] (* Alonso del Arte, Sep 06 2011 *)
LinearRecurrence[{3,-3,1},{1,10,34,74},50] (* Harvey P. Dale, Feb 13 2022 *)

vector(51, n, if(n==1,1, 2*(1+(2*n-2)^2)) ) \\ G. C. Greubel, Nov 09 2019

[1]+[2*(1+4*n^2) for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(0) = 1; a(n) = 8*n^2+2 for n>0.
From Gary W. Adamson, Dec 27 2007: (Start)
a(n) = (2n+1)^2 + (2n-1)^2 for n>0.
Binomial transform of [1, 9, 15, 1, -1, 1, -1, 1, ...]. (End)
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: (1+x)*(1+6*x+x^2)/(1-x)^3. (End)
From Bruce J. Nicholson, Jul 31 2019: (Start) Assume n>0.
a(n) = A016754(n) + A016754(n-1).
a(n) = 2 * A053755(n).
a(n) = A054554(n+1) + A054569(n+1).
a(n) = A033951(n) + A054552(n).
a(n) = A054556(n+1) + A054567(n+1). (End)
E.g.f.: -1 + 2*exp(x)*(1 + 2*x)^2. - Stefano Spezia, Aug 02 2019
Sum_{n>=0} 1/a(n) = 3/4+1/8*Pi*coth(Pi/2) = 1.178172.... - R. J. Mathar, May 07 2024

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

Original entry on oeis.org

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

cp's OEIS Frontend

A008527 Coordination sequence for body-centered tetragonal lattice.

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