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A161595
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The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.
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4
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1, 16, 271, 4591, 77776, 1317601, 22321441, 378146896, 6406175791, 108526841551, 1838550130576, 31146825378241, 527657481299521, 8939030356713616, 151435858582831951, 2565470565551429551, 43461563755791470416, 736281113282903567521, 12473317362053569177441
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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- 285*y^2=1,
with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
Also: the first differences of A078366.
Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - Colin Barker, Feb 14 2014
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LINKS
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FORMULA
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a(t+2) = 17*a(t+1)-a(t).
a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).
a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.
G.f.: x*(1-x)/(1-17*x+x^2).
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MAPLE
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t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t, a, b, n): end if: end do:
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MATHEMATICA
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Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2), {x, 0, 40}], x]] (* or *) LinearRecurrence[{17, -1}, {1, 16}, 20] (* Harvey P. Dale, Oct 12 2012 *)
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PROG
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(PARI) Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ Colin Barker, Feb 14 2014
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CROSSREFS
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Cf. similar sequences listed in A238379.
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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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