A055776 a(n) = a(n-1)^3 + a(n-1)^2 + a(n-1) + 1.

0, 1, 4, 85, 621436, 239988219843053389, 13821964488793901254190711941736196403535171578341580

Offset: 0

Author

Henry Bottomley, Jul 12 2000

Keywords

• nonn

Comments

The next term has 157 digits. - Harvey P. Dale, Dec 08 2019

Examples

			a(3) = 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.
		

References

• Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203

Links

• Wikipedia, Herbrand Structure
• Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.

Crossrefs

Cf. A002065.

Programs

Magma

[n le 1 select 0 else Self(n-1)^3 + Self(n-1)^2 + Self(n-1) + 1: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015

Mathematica

RecurrenceTable[{a[n] == a[n - 1]^3 + a[n - 1]^2 + a[n - 1] + 1, a[0] == 0}, a, {n, 0, 6}] (* Michael De Vlieger, Oct 05 2015 *)
NestList[#^3+#^2+#+1&,0,7] (* Harvey P. Dale, Dec 08 2019 *)

PARI

a=vector(6);a[1]=1;print1("0, 1, ");for(n=2,6,a[n]=a[n-1]^3+a[n-1]^2+a[n-1]+1;print1(a[n],", ")) \\ Gerald McGarvey, Dec 08 2007

PARI

a(n) = if(n==0, 0, a(n-1)^3 + a(n-1)^2 + a(n-1) + 1);
vector(10, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

Formula

a(n) is asymptotic to c^(3^(n+1)) where c=1.056431004248312118265251254776175173104598976924006344252579493163876246969557582... - Gerald McGarvey, Dec 08 2007, corrected by Vaclav Kotesovec, Apr 03 2016

a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015

Extensions

Next term is too big to include.