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A161852
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Solutions to the simultaneous equations m(n)+1=a(n)^2 and 7*m(n)+1=b(n)^2.
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2
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1, 5, 11, 79, 175, 1259, 2789, 20065, 44449, 319781, 708395, 5096431, 11289871, 81223115, 179929541, 1294473409, 2867582785, 20630351429, 45701395019, 328791149455, 728354737519, 5240028039851, 11607974405285, 83511657488161, 184999235747041, 1330946491770725
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The equations are equivalent to the Pell equation x(n)^2-7*y(n)^2=9
with x(n)=7*m(n)+4 and y(n)=a(n)*b(n);
x-values in the solution to 7x^2 - 6 = y^2
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (16,0,-1).
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FORMULA
| Contribution from Bruno Berselli, Oct 28 2011: (Start)
G.f.: x*(1-x)*(1+6*x+x^2)/(1-16*x^2+x^4).
a(n) = ((7+(-1)^n*t)*(8-3*t)^floor(n/2)+(7-(-1)^n*t)*(8+3*t)^floor(n/2))/14 with t=sqrt(7). (End)
a(n) = 16*a(n-2) - a(n-4) with a(1)=1, a(2)=5, a(3)=11, a(4)=79. - Sture Sjöstedt, Nov 18 2011
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MAPLE
| n=0: for a from 1 to 1000000 do b:=sqrt(7*a^2-6):
if (trunc(b)=b) then n:=n+1: m:=a^2-1: x:=7*m+4: y:=a*b:
print(n, a, b, m, x, y): end if: end do:
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PROG
| (Maxima) makelist(expand(((7+(-1)^n*sqrt(7))*(8-3*sqrt(7))^floor(n/2)+(7-(-1)^n*sqrt(7))*(8+3*sqrt(7))^floor(n/2))/14), n, 1, 26); [Bruno Berselli, Oct 28 2011]
(PARI) Vec((1-x)*(1+6*x+x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 28 2011
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CROSSREFS
| Cf. A195878.
Sequence in context: A067890 A192436 A154797 * A002359 A090518 A057726
Adjacent sequences: A161849 A161850 A161851 * A161853 A161854 A161855
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KEYWORD
| nonn,easy
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AUTHOR
| Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 20 2009
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EXTENSIONS
| More terms from Bruno Berselli, Oct 28 2011
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