login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..15.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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