A067686 a(n) = a(n-1) * a(n-1) - B * a(n-1) + B, a(0) = 1 + B for B = 7.

8, 15, 127, 15247, 232364287, 53993160246468367, 2915261353400811631533974206368127, 8498748758632331927648392184620600167779995785955324343380396911247

Offset: 0

Author

Drastich Stanislav (drass(AT)spas.sk), Feb 05 2002

Keywords

• nonn
• easy

Comments

This is the special case k=7 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
• S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
• S. Mustonen, On integer sequences with mutual k-residues
• Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
• Index entries for sequences of form a(n+1)=a(n)^2 + ....

Crossrefs

Cf. B=1: A000058 (Sylvester's sequence), B=2: A000215 (Fermat numbers), B=3: A000289, B=4: A000324, B=5: A001543, B=6: A001544.

Column k=7 of A177888.

Programs

Mathematica

RecurrenceTable[{a[0]==8, a[n]==a[n-1]*(a[n-1]-7)+7}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)
NestList[#^2-7#+7&,8,10] (* Harvey P. Dale, Jan 26 2025 *)

Formula

a(n) ~ c^(2^n), where c = 3.3333858371760195832345950846454963835549715770476958790043961891683146201... . - Vaclav Kotesovec, Dec 17 2014