A206526 a(n) = 137*(n-1) - a(n-1) with n>1, a(1)=31.

31, 106, 168, 243, 305, 380, 442, 517, 579, 654, 716, 791, 853, 928, 990, 1065, 1127, 1202, 1264, 1339, 1401, 1476, 1538, 1613, 1675, 1750, 1812, 1887, 1949, 2024, 2086, 2161, 2223, 2298, 2360, 2435, 2497, 2572, 2634, 2709, 2771, 2846, 2908, 2983

Offset: 1

Author

Vincenzo Librandi, Mar 09 2012

Keywords

• nonn
• easy

Comments

Positive numbers k such that k^2 == 2 (mod 137), where the prime 137 == 1 (mod 8).

Equivalently, numbers k such that k == 31 or 106 (mod 137).

The subsequence of primes begins: 31, 853, 1613, 1949, 2161. - Jonathan Vos Post, Mar 09 2012

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, A206525.

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

[(-137+13*(-1)^n+274*n)/4: n in [1..60]];

Magma

[n: n in [1..3000] | n^2 mod 137 eq 2]; // Vincenzo Librandi, Mar 31 2016

Mathematica

LinearRecurrence[{1, 1, -1}, {31, 106, 168}, 40] (* or *) CoefficientList[Series[x*(31+75*x+31*x^2)/((1+x)*(x-1)^2), {x, 0, 50}], x] (* or *) a[1] = 31; a[n_] := a[n] = 137*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]

Formula

a(n) = a(n-2) + 137.

G.f.: x*(31+75*x+31*x^2)/((1+x)*(x-1)^2).

a(n) = (-137+13*(-1)^n+274*n)/4.

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

Sum_{n>=1} (-1)^(n+1)/a(n) = cot(31*Pi/137)*Pi/137. - Amiram Eldar, Feb 28 2023