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A299647
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Positive solutions to x^2 == -2 (mod 11).
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1
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3, 8, 14, 19, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 80, 85, 91, 96, 102, 107, 113, 118, 124, 129, 135, 140, 146, 151, 157, 162, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 278, 283, 289, 294, 300, 305, 311, 316
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OFFSET
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1,1
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COMMENTS
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Positive numbers congruent to {3, 8} mod 11.
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LINKS
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FORMULA
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O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - David Lovler, Aug 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
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MATHEMATICA
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Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
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PROG
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(PARI) vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)
(Sage) [5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]
(Maxima) makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);
(GAP) List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);
(Magma) [5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];
(Python) [5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]
(Julia) [(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018
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CROSSREFS
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Cf. A017497: positive solutions to x == -2 (mod 11).
Cf. A017437: positive solutions to x^3 == -2 (mod 11).
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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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