**Want 3 consecutive integers?**

**Take 4 consecutive terms!**

Hello SeqFans,

I must have missed
the correct entry in the OEIS because I cannot find this easy (core?) seq.

"To find 3
consecutive naturals in S, you have to take 4 consecutive terms of S -- no
less":

S =
1,3,0,2,4,1,3,5,2,4,6,3,5,7,4,6,8,5,7,9,6,8,10,7,9,11,8,10,12,9,11,13,10,12,14,11,13,15,12,14,16,13,15,17,14,16,18,15,...

Ex: taking the
first 3 terms doesn’t allow you to handle 3 consecutive natural numbers as they
are 0,1 and... 3.

But if you take
the fourth term (2), you’ll have in hand 0,1,2 [and
even another triplet of consecutive naturals, which is (1,2,3)].

Formula is easy to
compute.

Best,

E.

__________

**Alex M**.

Simplest formula I
could get: a(3n)=n-1; a(3n+1)=n+1; a(3n+2)=n+3. Or, a(n)=1+a(n-3).

__________

**Richard Mathar**:

Confirming Alex we
see that this falls into the pattern

a(n)= +a(n-1)
+a(n-3) -a(n-4) = (n+3+5*A049347(n-1))/3
(assuming offset 0)

also
known as the ocean of

<a href="Sindx_Rea.html#recLCC">Index
to sequences with linear recurrences with constant coefficients</a>,
signature (1,0,1,-1).

Generating
function

(1+2*x+x^3-3*x^2)/(1+x+x^2)/(x-1)^2

These
almost-no-growth sequences are bad for the economy.

**RJM**

__________

**Alexander P-sky**:

PURRS Demo Results

Verified exact
solution for x(n) = 1+x(-3+n) for the initial
conditions

x(0) = 1

x(1) = 3

x(2) = 0

Verified solution x(n) =

1+1/3*n-(1/18*I)*sqrt(3)*(-1/2+(1/2*I)*sqrt(3))^n+1/2*(-1/2-(1/2*I)*sqrt(3))^n+1

/2*(-1/2+(1/2*I)*sqrt(3))^n+(1/18*I)*sqrt(3)*(-1/2-(1/2*I)*sqrt(3))^n+(-1/2-(1/2

*I)*sqrt(3))^(-1)*(-1/2-(1/2*I)*sqrt(3))^n+(-1/2+(1/2*I)*sqrt(3))^(-1)*(-1/2+(1

/2*I)*sqrt(3))^n

for each n >= 0

---

PURRS Demo Results

Verified exact
solution for x(n) = x(-1+n)-x(-4+n)+x(-3+n) for the
initial conditions

x(0) = 1

x(1) = 3

x(2) = 0

x(3) = 2

Verified solution

x(n) =
1+1/3*n-(5/9*I)*sqrt(3)*(-1/2+(1/2*I)*sqrt(3))^n+(5/9*I)*sqrt(3)*(-1/2-(1/2*I)*sqrt(3))^n

for each n >= 0

Computing the
exact solution took about 61 ms of CPU time; verifying it took about 19 ms of
CPU time.

__________

S building method
is easy:

S = 1,3,0,2,4,1,3,5,2,4,6,3,5,7,4,6,8,5,7,9,6,8,10,7,9,11,8,10,12,9,11,13,10,12,14,11,13,15,12,14,16,13,15,17,14,16,18,15,...

S = 1,3,0,2,4,1,3,5,2,4,6,3,5,7,4,6,8,5,7,9,6,8,10,7,9,11,8,10,12,9,11,13,10,12,14,11,13,15,12,14,16,13,15,17,14,16,18,15,...

Best,

Thanks to all,

É.