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A161584
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The list of the k values in the common solutions to the 2 equations 13*k+1=A^2, 17*k+1=B^2.
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2
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0, 15, 3360, 749280, 167086095, 37259449920, 8308690246080, 1852800665425935, 413166239699737440, 92134218652376023200, 20545517593240153436175, 4581558289073901840243840, 1021666952945886870220940160
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OFFSET
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1,2
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COMMENTS
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The 2 equations are equivalent to the Pell equation x^2-221*y^2=1,
with x=(221*k+15)/2 and y= A*B/2, case C=13 of A160682.
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LINKS
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FORMULA
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k(t+3)=224*(k(t+2)-k(t+1))+k(t).
k(t)=((15+w)*((223+15*w)/2)^(t-1)+(15-w)*((223-15*w)/2)^(t-1))/442 where w=sqrt(221).
k(t) = floor of ((15+w)*((223+15*w)/2)^(t-1))/442;
G.f.: -15*x^2/((x-1)*(x^2-223*x+1)).
a(1)=0, a(2)=15, a(3)=3360, a(n)=224*a(n-1)-224*a(n-2)+a(n-3). - Harvey P. Dale, Nov 22 2013
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MAPLE
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t:=0: for n from 0 to 1000000 do a:=sqrt(13*n+1): b:=sqrt(17*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t, n, a, b): end if: end do:
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MATHEMATICA
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LinearRecurrence[{224, -224, 1}, {0, 15, 3360}, 20] (* Harvey P. Dale, Nov 22 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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