OFFSET
0,1
COMMENTS
Each term equals the sum of three eighth powers and also twice a perfect square: a(n)= 2*(n^8+14n^4*2^4+2^8)^2.
More generally, a(n,k) = 2*(n^8+14*n^4*k^4+k^8)^2 = x^8+y^8+z^8, where x=n^2-k^2; y=n^2+k^2; z=2*n*k.
REFERENCES
Robert Carmichael, Diophantine Analysis, Ed. 1915 by Mathematical Monographs, pages 96
LINKS
Rafael Parra Machío, Ecuaciones Diofánticas, Tema XI, page 19.
Rafael Parra Machío, Teoría de Números, Web Site.
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188, -12376,19448,-24310,24310,-19448,12376, -6188,2380,-680,136,-17,1).
FORMULA
a(n) = 2*(n^8+14*n^4*2^4+2^8)^2.
G.f.: ( -131072 +1765502*x -43513950*x^2 -649478930*x^3 -13701900430*x^4 -195088344234*x^5 -1536270678326*x^6 -6277763482330*x^7 -12900117572550*x^8 -12896931212230*x^9 -6280312570586*x^10 -1534648531254*x^11 -195899417770*x^12 -13389949070*x^13 -738607890*x^14 -25688158*x^15 -462722*x^16 ) / (x-1)^17. - R. J. Mathar, Jun 04 2011
EXAMPLE
MATHEMATICA
Table[2(m^8+14m^4n^4+n^8)^2, {m, 1, 10}]/. n -> 2
Table[(m^2-n^2)^8+(m^2+n^2)^8+(2*m*n), {m, 1, 10}]/. n -> 2
Table[{(m^2-2^2), (m^2+2^2), (2*m*2)}, {m, 1, 5}], (* triples x, y, z *)
Table[2(n^8+224n^4+256)^2, {n, 0, 20}] (* Harvey P. Dale, Jun 19 2011 *)
PROG
(PARI) a(n)=2*(n^4+4*n^3+8*n^2-16*n+16)^2*(n^4-4*n^3+8*n^2+16*n+16)^2 ; \\ Charles R Greathouse IV, May 19 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rafael Parra Machio, May 19 2011
EXTENSIONS
More terms from Harvey P. Dale, Jun 29 2011
STATUS
approved