A376024 a(0..4) = 1 and a(n) = (a(n-2)^2 + a(n-3)^2 + a(n-2)*(3*a(n-3) + a(n-4)) + a(n-1)*(a(n-3) - a(n-5)))/(a(n-4) + a(n-5)) for n > 4.

1, 1, 1, 1, 1, 3, 3, 11, 35, 83, 545, 2513, 13905, 152721, 1087873, 14651923, 238834051, 3135275371, 91466933731, 2155382231811, 63058059937761, 3261572372004353, 120654520736448833, 8395343248160222081, 661217270644238022305, 46110296193095128622723, 6786635441262507324649635

Offset: 0

Author

Thomas Scheuerle, Sep 06 2024

Keywords

• nonn

Comments

An example of how a Somos recurrence can be generalized such that proving its integrality looks more difficult in the first glance. In this example the Somos-4 recurrence b(n) = (b(n-1) * b(n-3) + b(n-2)^2) / b(n-4) was modified by substitution of b(n-k) with (a(n-k) + a(n-k-1)).

This sequence is not a divisibility sequence unlike Somos-4 sequences are.

Crossrefs

Cf. A006720, A097495 ( first 6 values coincidence with odd terms ).

Programs

PARI

a=vector(26); a[1]=a[2]=a[3]=a[4]=a[5]=1; for(n=6, #a, a[n]=(a[n-2]^2+a[n-3]^2+a[n-2]*(3*a[n-3]+a[n-4])+a[n-1]*(a[n-3]-a[n-5]))/(a[n-4]+a[n-5])); a

Formula

(a(n) + a(n+1))/2 = A006720(n).