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A036666
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Numbers k such that 5*k + 1 is a square.
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18
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0, 3, 7, 16, 24, 39, 51, 72, 88, 115, 135, 168, 192, 231, 259, 304, 336, 387, 423, 480, 520, 583, 627, 696, 744, 819, 871, 952, 1008, 1095, 1155, 1248, 1312, 1411, 1479, 1584, 1656, 1767, 1843, 1960, 2040, 2163, 2247, 2376, 2464, 2599, 2691
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OFFSET
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1,2
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COMMENTS
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Third differences are 4, -6, 8, -10, 12, -14, 16, -18, 20, -22, 24, -26, 28, ...
X values of solutions to the equation 5*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007
Also, numbers 5*i^2 + 2*i for integer i. The characteristic function is A205633(n). - Jason Kimberley, Nov 15 2012
Match the values a(n) with the squares 5k + 1 as follows:
3,....7,....16,....24,... .a, a, a, a,...
16,...36,....81,...121,... (base).
Then 1/5 in the matching base is equal to .a, a, a,...
Example: 1/5 in base 36 is equal to .7, 7, 7, 7...
Check: 7/36 + 7/36^2 = 259/1296 = .199845...; close to 1/5.
(End)
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LINKS
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FORMULA
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G.f.: x*(3 + 4*x + 3*x^2) / ((1 - x)*(1 - x^2)).
a(n) has the form ((5*m + 1)^2 - 1)/5 if n is odd; a(n) has the form ((5*m + 4)^2 - 1)/5 if n is even.
a(n) = n^2 + n + ceiling(n/2)^2, (with offset 0). - Gary Detlefs, Feb 23 2010
a(n) = (10*n*(n - 1)+(2*n - 1)*(-1)^n + 1)/8.
Sum_{n>=2} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/2.
Sum_{n>=2} (-1)^n/a(n) = 5*(log(5)-1)/4 - sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). (End)
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MAPLE
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seq(n^2+n+ceil(n/2)^2, n=0..46); # Gary Detlefs, Feb 23 2010
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MATHEMATICA
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(Select[ Range[121], Mod[ #, 5] == 1 || Mod[ #, 5] == 4 &]^2 - 1)/5 (* Robert G. Wilson v, Jun 23 2004 *)
Flatten[Position[5*Range[0, 3000]+1, _?(IntegerQ[Sqrt[#]]&)]]-1 (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 7, 16, 24}, 50] (* Harvey P. Dale, Feb 13 2018 *)
Accumulate[Table[n + LCM[n, 2], {n, 0, 121}]] (* Jon Maiga, Nov 28 2018 *)
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PROG
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(PARI) a(n)=n^2+n+ceil(n/2)^2
(Magma) [(n-1)^2+(n-1)+Ceiling((n-1)/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Jun 05 2014
(GAP) List([1..50], n->(10*n*(n-1)+(2*n-1)*(-1)^n+1)/8); # Muniru A Asiru, Nov 28 2018
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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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Better description and additional formula from Santi Spadaro, Jul 12 2001
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STATUS
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approved
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