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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

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).

Programs

GAP

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

Magma

[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);

Mathematica

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 *)

PARI

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

Sage

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

Formula

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