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A174398 Numbers that are congruent to {1, 4, 5, 8} mod 12. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)