This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008527 #66 May 07 2024 04:45:10
%S A008527 1,10,34,74,130,202,290,394,514,650,802,970,1154,1354,1570,1802,2050,
%T A008527 2314,2594,2890,3202,3530,3874,4234,4610,5002,5410,5834,6274,6730,
%U A008527 7202,7690,8194,8714,9250,9802,10370,10954,11554,12170,12802,13450,14114,14794,15490,16202,16930,17674
%N A008527 Coordination sequence for body-centered tetragonal lattice.
%C A008527 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 A008527 Vincenzo Librandi, <a href="/A008527/b008527.txt">Table of n, a(n) for n = 0..10000</a>
%H A008527 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 A008527 M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H A008527 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A008527 a(0) = 1; a(n) = 8*n^2+2 for n>0.
%F A008527 From _Gary W. Adamson_, Dec 27 2007: (Start)
%F A008527 a(n) = (2n+1)^2 + (2n-1)^2 for n>0.
%F A008527 Binomial transform of [1, 9, 15, 1, -1, 1, -1, 1, ...]. (End)
%F A008527 From _Colin Barker_, Apr 13 2012: (Start)
%F A008527 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
%F A008527 G.f.: (1+x)*(1+6*x+x^2)/(1-x)^3. (End)
%F A008527 From _Bruce J. Nicholson_, Jul 31 2019: (Start) Assume n>0.
%F A008527 a(n) = A016754(n) + A016754(n-1).
%F A008527 a(n) = 2 * A053755(n).
%F A008527 a(n) = A054554(n+1) + A054569(n+1).
%F A008527 a(n) = A033951(n) + A054552(n).
%F A008527 a(n) = A054556(n+1) + A054567(n+1). (End)
%F A008527 E.g.f.: -1 + 2*exp(x)*(1 + 2*x)^2. - _Stefano Spezia_, Aug 02 2019
%F A008527 Sum_{n>=0} 1/a(n) = 3/4+1/8*Pi*coth(Pi/2) = 1.178172.... - _R. J. Mathar_, May 07 2024
%p A008527 1, seq(8*k^2+2, k=1..50);
%t A008527 a[0]:= 1; a[n_]:= 8n^2 +2; Table[a[n], {n,0,50}] (* _Alonso del Arte_, Sep 06 2011 *)
%t A008527 LinearRecurrence[{3,-3,1},{1,10,34,74},50] (* _Harvey P. Dale_, Feb 13 2022 *)
%o A008527 (PARI) vector(51, n, if(n==1,1, 2*(1+(2*n-2)^2)) ) \\ _G. C. Greubel_, Nov 09 2019
%o A008527 (Magma) [1] cat [2*(1 + 4*n^2): n in [1..50]]; // _G. C. Greubel_, Nov 09 2019
%o A008527 (Sage) [1]+[2*(1+4*n^2) for n in (1..40)] # _G. C. Greubel_, Nov 09 2019
%o A008527 (GAP) Concatenation([1], List([1..40], n-> 2*(1+4*n^2) )); # _G. C. Greubel_, Nov 09 2019
%Y A008527 Apart from leading term, same as A108100.
%Y A008527 Cf. A206399.
%Y A008527 Cf. A016754 (SE), A054554 (NE), A054569 (SW), A053755 (NW), A033951 (S), A054552 (E), A054556 (N), A054567 (W) (Ulam spiral spokes).
%Y A008527 A143839 (SSE) + A143855 (ESE) = A143838 (SSW) + A143856 (ENE) = A143854 (WSW) + A143861 (NNE) = A143859 (WNW) + A143860 (NNW) = even bisection = a(2n) = A010021(n).
%K A008527 nonn,easy
%O A008527 0,2
%A A008527 _N. J. A. Sloane_