Search: a082651 -id:a082651
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A052454
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Positive integer values of k such that 10*k^2 - 9 is a square.
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+10
4
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1, 3, 13, 25, 111, 493, 949, 4215, 18721, 36037, 160059, 710905, 1368457, 6078027, 26995669, 51965329, 230804967, 1025124517, 1973314045, 8764510719, 38927735977, 74933968381, 332820602355, 1478228842609, 2845517484433, 12638418378771
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = 38*a(n-3)-a(n-6).
G.f.: x*(1-x)*(1+4*x+17*x^2+4*x^3+x^4)/(1-38*x^3+x^6). [Colin Barker, Jun 13 2012]
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EXAMPLE
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25 is a term of the sequence since 10*25^2-9 = 6241 = 79^2.
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MATHEMATICA
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LinearRecurrence[{0, 0, 38, 0, 0, -1}, {1, 3, 13, 25, 111, 493}, 50] (* Paolo Xausa, Mar 18 2024 *)
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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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A281064
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Values of x such that x^2 = 5*y^2 + 11, where x and y are positive integers.
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+10
1
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4, 16, 56, 284, 1004, 5096, 18016, 91444, 323284, 1640896, 5801096, 29444684, 104096444, 528363416, 1867934896, 9481096804, 33518731684, 170131379056, 601469235416, 3052883726204, 10792927505804, 54781775692616, 193671225869056, 983019078740884
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OFFSET
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1,1
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COMMENTS
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The corresponding values of y are in A082651.
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LINKS
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FORMULA
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a(n) = ((-2-r)^n*(r-5) + (5+r)*(r-2)^n + (15+7*r)*(r+2)^n + (2-r)^n*(7*r-15)) / (4*r) where r=sqrt(5).
a(n) = 18*a(n-2) - a(n-4) for n>3.
G.f.: 4*x*(1 - x)*(1 + 5*x + x^2) / ((1 + 4*x - x^2)*(1 - 4*x - x^2)).
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EXAMPLE
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56 is in the sequence because 56^2 = 3136 = 5*25^2+11.
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MATHEMATICA
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LinearRecurrence[{0, 18, 0, -1}, {4, 16, 56, 284}, 30] (* Harvey P. Dale, May 28 2020 *)
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PROG
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(PARI) Vec(4*x*(1 - x)*(1 + 5*x + x^2) / ((1 + 4*x - x^2)*(1 - 4*x - x^2)) + O(x^30))
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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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STATUS
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approved
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