A206525 a(n) = 113*(n-1) - a(n-1) with n>1, a(1)=51.

51, 62, 164, 175, 277, 288, 390, 401, 503, 514, 616, 627, 729, 740, 842, 853, 955, 966, 1068, 1079, 1181, 1192, 1294, 1305, 1407, 1418, 1520, 1531, 1633, 1644, 1746, 1757, 1859, 1870, 1972, 1983, 2085, 2096, 2198, 2209

Offset: 1

Author

Vincenzo Librandi, Mar 09 2012

Keywords

• nonn
• easy

Comments

Positive numbers k such that k^2 == 2 (mod 113), where the prime 113 == 1 (mod 8).

Equivalently, numbers k such that k == 51 or 62 (mod 113).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010.

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204769.

Programs

Magma

[(-113-91*(-1)^n+226*n)/4: n in [1..60]];

Mathematica

LinearRecurrence[{1, 1, -1}, {51, 62, 164}, 40] (* or *) CoefficientList[Series[x*(51+11*x+51*x^2)/((1+x)*(x-1)^2), {x, 0, 40}], x] (* or *) a[1] = 51; a[n_] := a[n] = 113*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]

Formula

a(n) = a(n-2) + 113.

G.f.: x*(51+11*x+51*x^2)/((1+x)*(x-1)^2).

a(n) = (-113-91*(-1)^n+226*n)/4.

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

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(11*Pi/226)*Pi/113. - Amiram Eldar, Feb 28 2023