A174398 Numbers that are congruent to {1, 4, 5, 8} mod 12.

1, 4, 5, 8, 13, 16, 17, 20, 25, 28, 29, 32, 37, 40, 41, 44, 49, 52, 53, 56, 61, 64, 65, 68, 73, 76, 77, 80, 85, 88, 89, 92, 97, 100, 101, 104, 109, 112, 113, 116, 121, 124, 125, 128, 133, 136, 137, 140, 145, 148, 149, 152, 157, 160, 161, 164, 169, 172, 173

Offset: 1

Author

Gary Detlefs, Mar 18 2010

Keywords

• nonn
• easy

Comments

Numbers k such that k*(k + 3)/4 + (k + 1)*(k + 2)/6 or k*(5*k + 3)/12 + 1/3 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A087445, A092259.

Programs

Magma

[n : n in [0..200] | n mod 12 in [1, 4, 5, 8]]; // Wesley Ivan Hurt, Jun 07 2016

Maple

seq(3*n +(-1)^floor(n/2), n=0..50);

Mathematica

Table[(1+I)*(3*(n-n*I+I-1)+I^(1-n)-I^n)/2, {n, 60}] (* Wesley Ivan Hurt, Jun 07 2016 *)
Select[Range[200],MemberQ[{1,4,5,8},Mod[#,12]]&] (* or *) LinearRecurrence[ {2,-2,2,-1},{1,4,5,8},60] (* Harvey P. Dale, Aug 02 2020 *)

Formula

a(n) = 3*n - 3 + (-1)^floor((n-1)/2).

From Wesley Ivan Hurt, Jun 07 2016: (Start)

G.f.: x*(1 + 2*x - x^2 + 4*x^3)/((1 - x)^2*(1 + x^2)).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(n) = (1 + i)*(3*(n - n*i + i - 1) + i^(1-n) - i^n)/2, where i=sqrt(-1).

a(2*k) = A092259(k), a(2*k-1) = A087445(k). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 + log(2)/2. - Amiram Eldar, Dec 31 2021