A007204 Crystal ball sequence for D_4 lattice. (Formerly M5182)

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

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