A329429 -id:A329429 - cp's OEIS frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026

Offset: 0

N. J. A. Sloane and Richard Stanley

Number of binary trees of height less than or equal to n. [Corrected by Orson R. L. Peters, Jan 03 2020]
The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). - Jonathan Vos Post, Jul 21 2005
Apart from the initial term, a subsequence of A008318. - Reinhard Zumkeller, Jan 17 2008
Partial sums of A001699. - Jonathan Vos Post, Feb 17 2010
Corresponds to the second and second last diagonals of A119687. - John M. Campbell, Jul 25 2011
This is a divisibility sequence. - Michael Somos, Jan 01 2013
Sum_{n>=1} 1/a(n) = 1.739940825174794649210636285335916041018367182486941... . - Vaclav Kotesovec, Jan 30 2015
From Vladimir Vesic, Oct 03 2015: (Start)
Forming Herbrand's domains of formula: (∃x)(∀y)(∀z)(∃k)(P(x)∨Q(y)∧R(k))
where: x->a
       k->f(y,z)
we get:
  H0 = {a}
  H1 = {a, f(a,a)}
  H2 = {a, f(a,a), f(a,f(a,a)), f(f(a,a),a), f(f(a,a),f(a,a))}
  ...
The number of elements in each domain follows this sequence.
(End)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
This is a strong divisibility sequence; see A329429. - Clark Kimberling, Nov 13 2019
From Peter Bala, Oct 31 2022: (Start)
Let k be a positive integer. Clearly, the sequence obtained by reducing a(n) modulo k is eventually periodic. Conjectures:
1) The sequence obtained by reducing a(n) modulo 2^k is eventually periodic with period 2.
2) The sequence obtained by reducing a(n) modulo 10^k is eventually periodic with period 6 (the case k = 1 is noted above).
3) The sequence obtained by reducing a(n) modulo 20^k is eventually periodic with period 6.
4) For n >= floor(k/2) and for 1 <= i <= 6, the value of a(6*n+i) mod 10^k is a constant independent of n. The digits of these 6 constant integers, when read from right to left, are the first k digits of the 10-adic numbers A318135 (i = 1), A318136 (i = 2), A318137 (i = 3), A318138 (i = 4), A318139 (i = 5) and A318140 (i = 6), respectively. An example is given below.
  n      a(6*n+1) mod 10^11
1           10066388901
2           72084948901
3           67988948901
4           61588948901
5           01588948901
6           01588948901
7           01588948901
...             ...
A318135 begins 1, 0, 9, 8, 4, 9, 8, 8, 5, 1, 0, 2, .... (End)

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.
R. Penrose, The Emperor's New Mind, Oxford, 1989, p. 122.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..13
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!)
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1, Article 1.
P. Flajolet and A. M. Odlyzko, Limit distributions of coefficients of iterates of polynomials with applications to combinatorial enumerations, Math. Proc. Camb. Phil. Soc., 96 (1984), 237-253.
Claudio Gentile, Fabio Vitale, and Anand Rajagopalan, Flattening a Hierarchical Clustering through Active Learning, arXiv:1906.09458 [cs.LG], 2019.
Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of z^d + c, arXiv:1303.6513 [math.NT], 2013; to appear, Int. Math. Res. Not., c. 2015.
Dimitur Krustev, Simple Programs on Binary Trees Testing and Decidable Equivalence, 2016.
Robin Lamarche-Perrin, An Information-theoretic Framework for the Lossy Compression of Link Streams, arXiv:1807.06874 [cs.DS], 2018.
R. Lamarche-Perrin, Y. Demazeau, and J.-M. Vincent, A Generic Algorithmic Framework to Solve Special Versions of the Set Partitioning Problem, Preprint 105, Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig, 2014.
C. Lenormand, Arbres et permutations II, see p. 6.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Michael Penn, a stylish proof that..., YouTube video, 2021.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
M. Tainiter, Algebraic approach to stopping variable problems: Representation theory and applications, J. Combinatorial Theory 9 1970 148-161.
P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.
Stephan Wagner and Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index to divisibility sequences
Index entries for sequences related to Benford's law

Cf. A001699, A004019, A038044, A056207, A076949, A077496, A091980, A143848.
Cf. A143849, A213437, A247981, A248218, A248219, A318135, A318136, A318137.
Cf. A318138, A318139, A318140, A355108.
Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. - Robert Munafo, Nov 03 2009

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
[A003095(n): n in [0..12]]; // G. C. Greubel, Nov 29 2022

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)^2+1) end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 11 2019

NestList[#^2+1&,0,10] (* Harvey P. Dale, Dec 17 2010 *)

{a(n) = if( n<1, 0, 1 + a(n-1)^2)}; /* Michael Somos, Jan 01 2013 */

def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
[A003095(n) for n in range(13)] # G. C. Greubel, Nov 29 2022

a(n) = B_{n-1}(1) where B_n(x) = 1 + x*B_{n-1}(x)^2 is the generating function of trees of height <= n.
a(n) is asymptotic to c^(2^n) where c=1.2259024435287485386279474959130085213... (see A076949). - Benoit Cloitre, Nov 27 2002
c = b^(1/4) where b is the constant in Bottomley's formula in A004019. a(n) appears very asymptotic to c^(2^n) - Sum_{k>=1} A088674(k)/(2*c^(2^n))^(2*k-1). - Gerald McGarvey, Nov 17 2007
a(n) = Sum_{i=1..n} A001699(i). - Jonathan Vos Post, Feb 17 2010
G.f. = x + 2*x^2 + 5*x^3 + 26*x^4 + 677*x^5 + 458330*x^6 + 210066388901*x^7 + ... . - Michael Somos, Jan 01 2013
a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015
a(n) + a(n-1) = A213437(n). - Peter Bala, Feb 03 2017
0 = a(n)^2*(+a(n+1) + a(n+2)) + a(n+1)^2*(-a(n+1) - a(n+2) - a(n+3)) + a(n+2)^3 for all n>=0. - Michael Somos, Feb 10 2017
a(n) = A091980(2^(n-1)) for n > 0. - Alois P. Heinz, Jul 11 2019

Additional comments from Cyril Banderier, Jun 05 2000
Minor edits by Vaclav Kotesovec, Oct 04 2014
Initial term clarified by Clark Kimberling, Nov 13 2019

A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 19, 96, 224, 256, 128, 723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768, 1045459, 21100032, 209001984, 1339772928, 6194997248, 21845442560, 60641837056, 134967984128, 243130040320, 355391766528, 419950493696, 396881821696

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = 2 x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 1, 3, 19, 723, 1045459, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
  1;
  1, 2;
  3, 8, 8;
  19, 96, 224, 256, 128;
  723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + 2 x^2
  3 + 8 x^2 + 8 x^4
  19 + 96 x^2 + 224 x^4 + 256 x^6 + 128 x^8
  723 + 7296 x^2 + 35456 x^4 + 105472 x^6 + 208384 x^8 + 278528 x^10 + 245760 x^12 + 131072 x^14 + 32768 x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329431, A329433.

f[x_] := 2 x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329432 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329432 array *)

A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 9, 36, 90, 147, 171, 144, 87, 36, 9, 1, 730, 8748, 56862, 257337, 895941, 2528172, 5967108, 12025098, 20984508, 32024268, 43036029, 51168267, 53983503, 50609772, 42164064, 31176036, 20403009, 11768247, 5946156, 2610171, 984420, 314262, 83214, 17766, 2934, 351, 27, 1

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = x^3 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).

Then the sequence (p(n,0)) = (1,1,2,9,730, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..3:
  1;
  1, 1;
  2, 3, 3, 1;
  9, 36, 90, 147, 171, 144, 87, 36, 9, 1.
Rows 0..3, the polynomials u(n,x):
  1;
  1 + x^3;
  2 + 3 x^3 + 3 x^6 + x^9;
  9 + 36 x^3 + 90 x^6 + 147 x^9 + 171 x^12 + 144 x^15 + 87 x^18 + 36 x^21 + 9 x^24 + x^27.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329431, A329432, A329433.

f[x_] := x^3 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329430 polynomials u(n,x) *)
Table[CoefficientList[u[n, x^(1/3)], x], {n, 0, 5}]  (* A329430 array *)

A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 38, 48, 28, 8, 1, 1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1, 2090918, 10550016, 26125248, 41867904, 48398416, 42666880, 29610272, 16475584, 7419740, 2711424, 800992, 189248, 35064, 4928, 496, 32, 1, 4371938082726, 44118436709376

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).

Then the sequence (p(n,0)) = (1,2,6,38,1446, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889. p(n,0) = A072191(n) for n >= 1.

			Rows 0..4:
  1;
  2, 1;
  6, 4, 1;
  38, 48, 28, 8, 1;
  1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + x^2;
  6 + 4 x^2 + x^4;
  38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;
  1446 + 3648 x^2 + 4432 x^4 + 3296 x^6 + 1628 x^8 + 544 x^10 + 120 x^12 + 16 x^14 + x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329432, A329433.

f[x_] := x^2 + 2;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329431 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329431 array *)

A329433 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 147, 144, 60, 12, 1, 21612, 42336, 38376, 20808, 7350, 1728, 264, 24, 1, 467078547, 1829931264, 3451101120, 4148777664, 3552268752, 2294085888, 1154824416, 461895840, 148272828, 38314944, 7942320, 1306800, 167340, 16128, 1104, 48, 1

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = x^2 + 3, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 3, 12, 147, 21612, 467078547,... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
  1;
  3, 1;
  12, 6, 1;
  147, 144, 60, 12, 1;
  21612, 42336, 38376, 20808, 7350, 1728, 264, 24, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  3 + x^2;
  12 + 6 x^2 + x^4;
  147 + 144 x^2 + 60 x^4 + 12 x^6 + x^8;
  21612 + 42336 x^2 + 38376 x^4 + 20808 x^6 + 7350 x^8 + 1728 x^10 + 264 x^12 + 24 x^14 + x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329431, A329432.

f[x_] := x^2 + 3;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329433 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329433 array *)

A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 2, 21, 24, 8, 885, 2016, 1824, 768, 128, 1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768, 4907550002421, 44716844551680, 193253086462464, 525562214510592, 1006302608418816, 1438003249348608, 1586056913289216

Offset: 0

Clark Kimberling, Dec 07 2019

Let f(x) = 2 x^2 + 3, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 3, 21, 885, 1566453, 4907550002421, 48168094052524714211722485, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
  1;
  3, 2;
  21, 24, 8;
  885, 2016, 1824, 768, 128;
  1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768.
Rows 0..4, the polynomials u(n,x):
  1;
  3 + 2 x^2;
  21 + 24 x^2 + 8 x^4;
  885 + 2016 x^2 + 1824 x^4 + 768 x^6 + 128 x^8;
  1566453 + 7136640 x^2 + 14585472 x^4 + 17427456 x^6 + 13300224 x^8 + 6635520 + x^10 + 2113536 x^12 + 393216 x^14 +
  32768 x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329431, A329432, A329442.

f[x_] := 2 x^2 + 3;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329441 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329441 array *)

A329442 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 3, 14, 36, 27, 590, 3024, 6156, 5832, 2187, 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907, 3271700001614, 67075266827520, 652229166810816, 3990988066439808, 17193623473530864, 55281675697126272

Offset: 0

Clark Kimberling, Dec 07 2019

Let f(x) = 3 x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 2, 14, 590, 1044302, 3271700001614, ...) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
  1;
  2, 3;
  14, 36, 27;
  590, 3024, 6156, 5832, 2187;
  1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + 3 x^2;
  14 + 36 x^2 + 27 x^4;
  590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
  1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
  x^14 + 14348907 x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329431, A329432, A329441.

f[x_] := 3 x^2 + 2;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329442 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329442 array *)

cp's OEIS Frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

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A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329433 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329442 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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Mathematica