A067686 - OEIS

%I #30 Jan 26 2025 19:00:37

%S 8,15,127,15247,232364287,53993160246468367,

%T 2915261353400811631533974206368127,

%U 8498748758632331927648392184620600167779995785955324343380396911247

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

%C 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

%H Alois P. Heinz, <a href="/A067686/b067686.txt">Table of n, a(n) for n = 0..10</a>

%H A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.

%H A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

%H Stanislav Drastich, <a href="http://arxiv.org/abs/math/0202010">Rapid growth sequences</a>, arXiv:math/0202010 [math.GM], 2002.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2311857">On certain nonlinear recurring sequences</a>, Amer. Math. Monthly 70 (1963), 403-405.

%H S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>

%H Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a> [Local copy]

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>.

%F a(n) ~ c^(2^n), where c = 3.3333858371760195832345950846454963835549715770476958790043961891683146201... . - _Vaclav Kotesovec_, Dec 17 2014

%t RecurrenceTable[{a[0]==8, a[n]==a[n-1]*(a[n-1]-7)+7}, a, {n, 0, 10}] (* _Vaclav Kotesovec_, Dec 17 2014 *)

%t NestList[#^2-7#+7&,8,10] (* _Harvey P. Dale_, Jan 26 2025 *)

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

%Y Column k=7 of A177888.

%K nonn,easy,changed

%O 0,1

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