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A299647 Positive solutions to x^2 == -2 (mod 11). 1

%I

%S 3,8,14,19,25,30,36,41,47,52,58,63,69,74,80,85,91,96,102,107,113,118,

%T 124,129,135,140,146,151,157,162,168,173,179,184,190,195,201,206,212,

%U 217,223,228,234,239,245,250,256,261,267,272,278,283,289,294,300,305,311,316

%N Positive solutions to x^2 == -2 (mod 11).

%C Positive numbers congruent to {3, 8} mod 11. Equivalently, interleaving of A017425 and A017485.

%H Colin Barker, <a href="/A299647/b299647.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).

%F E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.

%F a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).

%F a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.

%F a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).

%F a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.

%F a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.

%t Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]

%t CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)

%t LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* _Robert G. Wilson v_, Mar 08 2018 *)

%o (PARI) vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)

%o (Sage) [5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]

%o (Maxima) makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);

%o (GAP) List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);

%o (MAGMA) [5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];

%o (Python) [5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]

%o (Julia) [(11(2n-1)-(-1)^n)>>2 for n in 1:60] # _Peter Luschny_, Mar 07 2018

%Y Subsequence of A106252, A279000.

%Y Cf. A017425, A017485.

%Y Cf. A017497: positive solutions to x == -2 (mod 11).

%Y Cf. A017437: positive solutions to x^3 == -2 (mod 11).

%Y Nonnegative solutions to x^2 == -2 (mod j): A005843 (j=2), A001651 (j=3), A047235 (j=6), A156638 (j=9), this sequence (j=11).

%K nonn,easy

%O 1,1

%A _Bruno Berselli_, Mar 06 2018

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Last modified March 3 10:32 EST 2021. Contains 341761 sequences. (Running on oeis4.)