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A190780
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a(n) = 2*(n^8 + 224*n^4 + 256)^2.
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0
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131072, 462722, 33554432, 1246103042, 30324948992, 563669272322, 7763186941952, 79452617800322, 626224351281152, 3963462651845762, 20906139893891072, 94733225757031682, 377800938372595712, 1351791004705013762, 4406854039510188032, 13253329257388072322
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OFFSET
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0,1
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COMMENTS
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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.
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REFERENCES
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Robert Carmichael, Diophantine Analysis, Ed. 1915 by Mathematical Monographs, pages 96
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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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},
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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