# Is OEIS A248049 an integer sequence?

The OEIS sequence A248049 defined by

$$a_n \!=\! (a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})/a_{n-4} \;\text{ with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$

is apparently an integer sequence but I have no proofs. I have numerical evidence using PARI/GP and Mathematica only. It is a real problem because its companion OEIS sequence A248048 has the same recursion with $$\,a_0=-1, a_1=a_2=a_3=1\,$$ but now $$\,a_{144}\,$$ has a denominator of $$2$$. There is a resemblance to the Somos-4 sequence but that probably won't help with an integrality proof.

I have some interesting unproven observations about its factorization algebraically and $$p$$-adically for a few small values of $$p$$, but nothing that would prove integrality. For example, if $$\,x_0,x_1,x_2,x_3\,$$ are indeterminates, and we use initial values of $$a_0=x_0,\; a_1=x_1,\; a_2=x_2,\; a_3=x_3 \;\text{ and }\; x_4 := x_1+x_2,$$ with the same recursion, then $$\,a_n\,$$ has denominator a monomial in $$\,x_0,x_1,x_2,x_3,x_4\,$$ with exponents from OEIS sequence A023434. Since $$\,x_0=x_4=2\,$$ with the original sequence I can't prove that the numerator has enough powers of $$2$$ to compensate. Another example is that $$\,a_{12n+k}\,$$ is odd for $$\,k=1,2,3\,$$ and even for the other residue classes modulo $$12$$. I also have some further observations about its $$2$$-adic valuation behavior which I can't prove.

By the way, the sequence grows very fast. My best estimate is $$\,\log(a_n) \approx 1.25255\, c^n\,$$ where $$\,c\,$$ is the plastic constant OEIS sequence A060006. Note that $$x^4-x^3-x^2+1 = (x-1)(x^3-x-1)$$ and $$\,c\,$$ is the real root of the cubic factor.

Can anyone give a proof of integrality of A248049?

• If you have $p$-integrality for all $p$, then the (global) integrality follows. – WhatsUp Mar 19 at 3:13
• Reminds me of Somos Sequence's. – Vepir Mar 19 at 11:27
• @Vepir Probably not surprising because the author of OEIS sequence A248049 is Michael Somos. – user Mar 19 at 11:34
• Similar to math.stackexchange.com/questions/1905063/… (note: the proof is less than 100% verified and rather unsatisfactory in its brute-force component). – darij grinberg Mar 21 at 15:08