A001543 -id:A001543

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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.

+10
9

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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, alternative link.

EXAMPLE

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, ...

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 14 2010

STATUS

approved

A067686

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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, alternative link.

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 + ....

FORMULA

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

MATHEMATICA

RecurrenceTable[{a[0]==8, a[n]==a[n-1]*(a[n-1]-7)+7}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

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.

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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, alternative link.

FORMULA

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)}.

EXAMPLE

Triangle begins:

1, 1;

2, 1;

2, 4, 1;

4, 14, 16, 8, 1;

16, 112, 324, 508, 474, 268, 88, 16, 1;

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888. First column gives: A165420.

KEYWORD

nonn,tabf

AUTHOR

Vladimir Shevelev, Dec 11 2010

EXTENSIONS

More terms from Alois P. Heinz, Dec 13 2010

STATUS

approved

A275698

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..8.

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]

FORMULA

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).

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Andres Cicuttin, Aug 05 2016

STATUS

approved

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