A190780 a(n) = 2*(n^8 + 224*n^4 + 256)^2.

131072, 462722, 33554432, 1246103042, 30324948992, 563669272322, 7763186941952, 79452617800322, 626224351281152, 3963462651845762, 20906139893891072, 94733225757031682, 377800938372595712, 1351791004705013762, 4406854039510188032, 13253329257388072322

Offset: 0

Author

Rafael Parra Machio, May 19 2011

Keywords

• nonn
• easy

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.

Examples

			462722 = 3^8+5^8+4^8 = 2*481^2.
563669272322 = 21^8+29^8+20^8 = 2*481^2.
Triplets (x,y,z) for k=2: {-3,5,4}, {0,8,8}, {5,13,12}, {12,20,16}, {21,29,20}, {32,40,24}, {45,53,28}, {60, 68,32}, {77,85,36},
{96,104,40}, see A028347 for x, A087475 for y, A008586 for z.
		

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).

Programs

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 *)

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

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

Extensions

More terms from Harvey P. Dale, Jun 29 2011