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

1,2

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

Table of n, a(n) for n=1..59.

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

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

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 *)

Prog

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

Crossrefs

Cf. A087445, A092259.

Sequence in context: A297419 A230549 A133940 * A341420 A030978 A101948

Adjacent sequences: A174395 A174396 A174397 * A174399 A174400 A174401

Keyword

nonn,easy

Author

Gary Detlefs, Mar 18 2010

Status

approved