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

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

Offset

1,1

Comments

Positive numbers congruent to {3, 8} mod 11.

Equivalently, interleaving of A017425 and A017485.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

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

From Amiram Eldar, Nov 23 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.

Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)

Mathematica

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 *)
LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* Robert G. Wilson v, Mar 08 2018 *)

Prog

(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

Crossrefs

Subsequence of A106252, A279000.

Cf. A017425, A017485.

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

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

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

Sequence in context: A140492 A184517 A028252 * A063617 A062550 A219930

Adjacent sequences: A299644 A299645 A299646 * A299648 A299649 A299650

Keyword

nonn,easy

Author

Bruno Berselli, Mar 06 2018

Status

approved