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 A007204 Crystal ball sequence for D_4 lattice. (Formerly M5182) 11
 1, 25, 169, 625, 1681, 3721, 7225, 12769, 21025, 32761, 48841, 70225, 97969, 133225, 177241, 231361, 297025, 375769, 469225, 579121, 707281, 855625, 1026169, 1221025, 1442401, 1692601, 1974025, 2289169, 2640625, 3031081, 3463321, 3940225, 4464769, 5040025 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Equals binomial transform of [1, 24, 120, 192, 96, 0, 0, 0, ...]. - Gary W. Adamson, Aug 13 2009 Hypotenuse of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + A069074(n-1)^2 = a(n)^2. - Martin Renner, Nov 12 2011 Numbers n such that n*x^4 + x^2 + 1 is reducible. - Arkadiusz Wesolowski, Nov 04 2013 REFERENCES Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf). Index entries for crystal ball sequences Index entries for sequences related to D_4 lattice Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA G.f.: (1 + 54*x^2 + 20*x + 20*x^3 + x^4)/(1-x)^5. a(0)=1, a(1)=25, a(2)=169, a(3)=625, a(4)=1681, a(n)=5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Mar 03 2013 Sum_{n>=0} 1/a(n) = Pi*(sinh(Pi) - Pi)/(2*(cosh(Pi) + 1)) = 1.0487582722070177... - Ilya Gutkovskiy, Nov 18 2016 a(n) = A016754(n) + A060300(n). - Bruce J. Nicholson, Apr 14 2017 a(n) = A001844(n)^2. - Bruce J. Nicholson, May 15 2017 a(n) = A000583(n+1) + A099761(n) + 2*A005563(n-1)*A000290(n). - Charlie Marion, Jan 14 2021 E.g.f.: exp(x)*(1 + 24*x + 60*x^2 + 32*x^3 + 4*x^4). - Stefano Spezia, Jun 06 2021 MAPLE A007204:=n->(2*n^2+2*n+1)^2; seq(A007204(n), n=0..30); MATHEMATICA Table[(2n^2+2n+1)^2, {n, 0, 30}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 25, 169, 625, 1681}, 40] (* Harvey P. Dale, Mar 03 2013 *) PROG (Magma) [(2*n^2+2*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Nov 18 2016 (PARI) a(n)=(2*n^2+2*n+1)^2 \\ Charles R Greathouse IV, Feb 08 2017 CROSSREFS Cf. A016754, A057769, A060300, A069074. Cf. A000290, A000583, A001844, A005563, A099761. Sequence in context: A198436 A080109 A017126 * A120096 A115330 A213546 Adjacent sequences: A007201 A007202 A007203 * A007205 A007206 A007207 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane and J. H. Conway, Apr 28 1994 EXTENSIONS More terms from Harvey P. Dale, Mar 03 2013 STATUS approved

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Last modified December 5 13:14 EST 2023. Contains 367591 sequences. (Running on oeis4.)