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

Author

Bruno Berselli, Mar 06 2018

Keywords

• nonn
• easy

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).

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).

Programs

GAP

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

Julia

[(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018

Magma

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

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 *)

Maxima

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

PARI

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

Python

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

Sage

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

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)