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 A190780 a(n) = 2*(n^8 + 224*n^4 + 256)^2. 0
 131072, 462722, 33554432, 1246103042, 30324948992, 563669272322, 7763186941952, 79452617800322, 626224351281152, 3963462651845762, 20906139893891072, 94733225757031682, 377800938372595712, 1351791004705013762, 4406854039510188032, 13253329257388072322 (list; graph; refs; listen; history; text; internal format)
 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 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. 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 Sequence in context: A289479 A222529 A069278 * A017698 A010805 A138032 Adjacent sequences:  A190777 A190778 A190779 * A190781 A190782 A190783 KEYWORD nonn,easy AUTHOR Rafael Parra Machio, May 19 2011 EXTENSIONS More terms from Harvey P. Dale, Jun 29 2011 STATUS approved

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Last modified February 18 15:30 EST 2020. Contains 332019 sequences. (Running on oeis4.)