A007204 Crystal ball sequence for D_4 lattice.

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

Author

N. J. A. Sloane and J. H. Conway, Apr 28 1994

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A016754, A057769, A060300, A069074.

Cf. A000290, A000583, A001844, A005563, A099761.

Programs

Magma

[(2*n^2+2*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Nov 18 2016

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 *)

PARI

a(n)=(2*n^2+2*n+1)^2 \\ Charles R Greathouse IV, Feb 08 2017

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 = (2*n^2+2*n+1)^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

Extensions

More terms from Harvey P. Dale, Mar 03 2013