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A219257
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Numbers k such that 11*k+1 is a square.
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17
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0, 9, 13, 40, 48, 93, 105, 168, 184, 265, 285, 384, 408, 525, 553, 688, 720, 873, 909, 1080, 1120, 1309, 1353, 1560, 1608, 1833, 1885, 2128, 2184, 2445, 2505, 2784, 2848, 3145, 3213, 3528, 3600, 3933, 4009, 4360, 4440, 4809, 4893, 5280, 5368, 5773, 5865
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OFFSET
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1,2
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COMMENTS
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Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/11.
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LINKS
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FORMULA
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G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022
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MATHEMATICA
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Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
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PROG
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(Magma) [n: n in [0..7000] | IsSquare(11*n+1)];
(Magma) I:=[0, 9, 13, 40, 48]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
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CROSSREFS
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Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).
Cf. A175885 (square roots of 11*a(n)+1).
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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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