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A008527 Coordination sequence for body-centered tetragonal lattice. 34
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 (list; graph; refs; listen; history; text; internal format)
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
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]
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified February 28 18:11 EST 2024. Contains 370400 sequences. (Running on oeis4.)