A001544 -id:A001544 - cp's OEIS frontend

A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 17, 43, 1, 6, 9, 31, 257, 1807, 1, 7, 11, 49, 871, 65537, 3263443, 1, 8, 13, 71, 2209, 756031, 4294967297, 10650056950807, 1, 9, 15, 97, 4691, 4870849, 571580604871, 18446744073709551617, 113423713055421844361000443, 1

Offset: 0

Alois P. Heinz, Dec 14 2010

			Square array P_n(k) begins:
  1,              2,          3,      4,       5,    6,    7,     8, ...
  1,              3,          5,      7,       9,   11,   13,    15, ...
  1,              7,         17,     31,      49,   71,   97,   127, ...
  1,             43,        257,    871,    2209, 4691, 8833, 15247, ...
  1,           1807,      65537, 756031, 4870849,  ...
  1,        3263443, 4294967297,    ...
  1, 10650056950807,        ...

Alois P. Heinz, Antidiagonals n = 0..13, flattened
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!)

Columns k=0-10 give: A000012, A000058(n+1), A000215, A000289(n+1), A000324(n+1), A001543(n+1), A001544(n+1), A067686, A110360(n+1), A110368(n+1), A110383(n+1).
Rows n=0-2 give: A000027(k+1), A005408, A056220(k+1).
Main diagonal gives A252730.
Coefficients of P_n(z) give: A177701.

p:= proc(n) option remember;
      z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
seq(seq(p(n)(d-n), n=0..d), d=0..8);

p[n_] := p[n] = Function[z, z + If [n == 0, 1, p[n-1][z]*(p[n-1][z]-z)] ]; Table [Table[p[n][d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

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

Original entry on oeis.org

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

Offset: 0

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

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

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 + ....

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.

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 *)

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

A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 4, 14, 16, 8, 1, 16, 112, 324, 508, 474, 268, 88, 16, 1, 256, 3584, 22912, 88832, 233936, 443936, 628064, 675456, 557492, 353740, 171644, 62878, 17000, 3264, 416, 32, 1, 65536, 1835008, 24576000, 209715200, 1281482752, 5974786048, 22114709504, 66752724992

Offset: 1

Vladimir Shevelev, Dec 11 2010

Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.

			Triangle begins:
   1,   1;
   2,   1;
   2,   4,   1;
   4,  14,  16,   8,   1;
  16, 112, 324, 508, 474, 268, 88, 16, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035
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!)

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888.

First column gives: A165420.

p:= proc(n) option remember;
       z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
deg:= n-> `if`(n=0, 1, 2^(n-1)):
T:= (n,k)-> coeff(p(n)(z), z, deg(n)-k):
seq(seq(T(n,k), k=0..deg(n)), n=0..6); # Alois P. Heinz, Dec 13 2010

P[0][z_] := z + 1;
P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
row[n_] := CoefficientList[P[n][z], z] // Reverse;
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).

Private values: P_n(0)=1; P_n(-1)=delta_(n,0)-1; {P_n(1)}=A000058; {P_n(2)}=A000215; {P_n(3)}={A000289(n+1)}; {P_n(4)}={A000324(n+1)}; {P_n(5)}={A001543(n+1)}; {P_n(6)}={A001544(n+1)}; {P_n(7)}={A067686(n)}; {P_n(8)}={A110360(n)}; {P_0(n)}={A000027(n+1)}; {P_1(n)}={A005408(n)}; {P_2(n)}={A056220(n+1)}.

More terms from Alois P. Heinz, Dec 13 2010

A275698 a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

Original entry on oeis.org

2, 5, 13, 133, 17293, 298995973, 89398590973228813, 7992108067998667938125889533702533, 63873791370569400659097694858350356285036046451665934814399129508493

Offset: 0

Andres Cicuttin, Aug 05 2016

nonn

This sequence could be considered a particular case of a possible two-parameter family of sequences of the form: a(n) = k1 + lcm(a(0),a(1),..,a(n-1)), a(0) = k2, where in this case k1=3 and k2=2. With other choices of k1 and k2 it seems it is possible to generate other sequences such as
A129871 with k1 = 1 and k2 = 1,
A000058 with k1 = 1 and k2 = 2,
A082732 with k1 = 1 and k2 = 3,
A000215 with k1 = 2 and k2 = 3,
A000324 with k1 = 4 and k2 = 1,
A001543 with k1 = 5 and k2 = 1,
A001544 with k1 = 6 and k2 = 1,
A275664 with k1 = 2 and k2 = 2,
A000289 with k1 = 3 and k2 = 1.

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]

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A082732, A275664.

A275698 = {2}; Do[AppendTo[A275698, 3 + LCM@@A275698], {i, 9}]; A275698

a(n) = 3 + lcm(a(0), a(1), ..., a(n - 1)), a(0) = 2.
a(n) = 3 + a(n-1)*(a(n-1)-3), for n > 1. - Christian Krause, Oct 17 2023. Proof: Follows from associativity of lcm(...) and the fact that gcd(m,m+3)=1:
a(n)-3 = lcm(a(0),a(1),...,a(n-2),a(n-1))
       = lcm(lcm(a(0),a(1),...,a(n-2)),a(n-1))
       = lcm(a(n-1)-3,a(n-1))
       = (a(n-1)-3)*a(n-1).

cp's OEIS Frontend

A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

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

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A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

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A275698 a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

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