This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A299647 #42 Nov 24 2024 01:52:02
%S A299647 3,8,14,19,25,30,36,41,47,52,58,63,69,74,80,85,91,96,102,107,113,118,
%T A299647 124,129,135,140,146,151,157,162,168,173,179,184,190,195,201,206,212,
%U A299647 217,223,228,234,239,245,250,256,261,267,272,278,283,289,294,300,305,311,316
%N A299647 Positive solutions to x^2 == -2 (mod 11).
%C A299647 Positive numbers congruent to {3, 8} mod 11.
%C A299647 Equivalently, interleaving of A017425 and A017485.
%H A299647 Colin Barker, <a href="/A299647/b299647.txt">Table of n, a(n) for n = 1..1000</a>
%H A299647 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A299647 O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
%F A299647 E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
%F A299647 a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
%F A299647 a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
%F A299647 a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
%F A299647 a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
%F A299647 a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
%F A299647 E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - _David Lovler_, Aug 08 2022
%F A299647 Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - _Amiram Eldar_, Feb 27 2023
%F A299647 From _Amiram Eldar_, Nov 23 2024: (Start)
%F A299647 Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.
%F A299647 Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)
%t A299647 Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
%t A299647 CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
%t A299647 LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* _Robert G. Wilson v_, Mar 08 2018 *)
%o A299647 (PARI) vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)
%o A299647 (Sage) [5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]
%o A299647 (Maxima) makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);
%o A299647 (GAP) List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);
%o A299647 (Magma) [5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];
%o A299647 (Python) [5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]
%o A299647 (Julia) [(11(2n-1)-(-1)^n)>>2 for n in 1:60] # _Peter Luschny_, Mar 07 2018
%Y A299647 Subsequence of A106252, A279000.
%Y A299647 Cf. A017425, A017485.
%Y A299647 Cf. A017497: positive solutions to x == -2 (mod 11).
%Y A299647 Cf. A017437: positive solutions to x^3 == -2 (mod 11).
%Y A299647 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 A299647 nonn,easy
%O A299647 1,1
%A A299647 _Bruno Berselli_, Mar 06 2018