A008527 - OEIS

%I #66 May 07 2024 04:45:10

%S 1,10,34,74,130,202,290,394,514,650,802,970,1154,1354,1570,1802,2050,

%T 2314,2594,2890,3202,3530,3874,4234,4610,5002,5410,5834,6274,6730,

%U 7202,7690,8194,8714,9250,9802,10370,10954,11554,12170,12802,13450,14114,14794,15490,16202,16930,17674

%N Coordination sequence for body-centered tetragonal lattice.

%C 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

%H Vincenzo Librandi, <a href="/A008527/b008527.txt">Table of n, a(n) for n = 0..10000</a>

%H M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.

%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(0) = 1; a(n) = 8*n^2+2 for n>0.

%F From _Gary W. Adamson_, Dec 27 2007: (Start)

%F a(n) = (2n+1)^2 + (2n-1)^2 for n>0.

%F Binomial transform of [1, 9, 15, 1, -1, 1, -1, 1, ...]. (End)

%F From _Colin Barker_, Apr 13 2012: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.

%F G.f.: (1+x)*(1+6*x+x^2)/(1-x)^3. (End)

%F From _Bruce J. Nicholson_, Jul 31 2019: (Start) Assume n>0.

%F a(n) = A016754(n) + A016754(n-1).

%F a(n) = 2 * A053755(n).

%F a(n) = A054554(n+1) + A054569(n+1).

%F a(n) = A033951(n) + A054552(n).

%F a(n) = A054556(n+1) + A054567(n+1). (End)

%F E.g.f.: -1 + 2*exp(x)*(1 + 2*x)^2. - _Stefano Spezia_, Aug 02 2019

%F Sum_{n>=0} 1/a(n) = 3/4+1/8*Pi*coth(Pi/2) = 1.178172.... - _R. J. Mathar_, May 07 2024

%p 1, seq(8*k^2+2, k=1..50);

%t a[0]:= 1; a[n_]:= 8n^2 +2; Table[a[n], {n,0,50}] (* _Alonso del Arte_, Sep 06 2011 *)

%t LinearRecurrence[{3,-3,1},{1,10,34,74},50] (* _Harvey P. Dale_, Feb 13 2022 *)

%o (PARI) vector(51, n, if(n==1,1, 2*(1+(2*n-2)^2)) ) \\ _G. C. Greubel_, Nov 09 2019

%o (Magma) [1] cat [2*(1 + 4*n^2): n in [1..50]]; // _G. C. Greubel_, Nov 09 2019

%o (Sage) [1]+[2*(1+4*n^2) for n in (1..40)] # _G. C. Greubel_, Nov 09 2019

%o (GAP) Concatenation([1], List([1..40], n-> 2*(1+4*n^2) )); # _G. C. Greubel_, Nov 09 2019

%Y Apart from leading term, same as A108100.

%Y Cf. A206399.

%Y Cf. A016754 (SE), A054554 (NE), A054569 (SW), A053755 (NW), A033951 (S), A054552 (E), A054556 (N), A054567 (W) (Ulam spiral spokes).

%Y A143839 (SSE) + A143855 (ESE) = A143838 (SSW) + A143856 (ENE) = A143854 (WSW) + A143861 (NNE) = A143859 (WNW) + A143860 (NNW) = even bisection = a(2n) = A010021(n).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_