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

0,2

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).

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

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 *)

Prog

(PARI) vector(51, n, if(n==1, 1, 2*(1+(2*n-2)^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
(Sage) [1]+[2*(1+4*n^2) for n in (1..40)] # G. C. Greubel, Nov 09 2019
(GAP) Concatenation([1], List([1..40], n-> 2*(1+4*n^2) )); # G. C. Greubel, Nov 09 2019

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).

Sequence in context: A020495 A155486 A225276 * A366415 A007584 A218329

Adjacent sequences: A008524 A008525 A008526 * A008528 A008529 A008530

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved