A003479 -id:A003479 - cp's OEIS frontend

A003229 a(n) = a(n-1) + 2*a(n-3) with a(0)=a(1)=1, a(2)=3.

1, 1, 3, 5, 7, 13, 23, 37, 63, 109, 183, 309, 527, 893, 1511, 2565, 4351, 7373, 12503, 21205, 35951, 60957, 103367, 175269, 297183, 503917, 854455, 1448821, 2456655, 4165565, 7063207, 11976517, 20307647, 34434061, 58387095, 99002389

Offset: 0

N. J. A. Sloane

Equals eigensequence of an infinite lower triangular matrix with 1's in the main diagonal, 0's in the subdiagonal and 2's in the subsubdiagonal (the triangle in the lower section of A155761). - Gary W. Adamson, Jan 28 2009
The operation in the comment of Jan 28 2009 is equivalent to the INVERT transform of (1, 0, 2, 0, 0, 0, ...). - Gary W. Adamson, Jan 21 2017
For n>=1, a(n) equals the number of ternary words of length n-1 having at least 2 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links below for an earlier version.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
D. E. Daykin, Letter to N. J. A. Sloane, Dec 1973.
D. E. Daykin, Letter to N. J. A. Sloane, Mar 1974.
D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 1.
D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 2.
D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 3.
D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 4.
D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, reverse side of page 4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 417.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (1,0,2).

Cf. A077949, A077974. First differences of A003479. Partial sums of A052537. Equals |A077906(n)|+|A077906(n+1)|.

Cf. A155761, A003230.

a003229 n = a003229_list !! n
a003229_list = 1 : 1 : 3 : zipWith (+)
                           (map (* 2) a003229_list) (drop 2 a003229_list)
-- Reinhard Zumkeller, Jan 01 2014

I:=[1, 1, 3]; [n le 3 select I[n] else Self(n-1)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 12 2012

seq(add(binomial(n-2*k,k)*2^k,k=0..floor(n/3)),n=1..38); # Zerinvary Lajos, Apr 03 2007
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >=3)}, unlabeled]: seq(count(SeqSeqSeqL, size=n+4), n=0..35); # Zerinvary Lajos, Apr 04 2009
a := n -> `if`(n<3,[1,1,3][n+1], hypergeom([-n/3, -(1+n)/3, (1-n)/3], [-n/2, -(1+n)/2], -27/2)); seq(simplify(a(n)),n=0..35); # Peter Luschny, Mar 09 2015

LinearRecurrence[{1,0, 2},{1,1,3},40] (* Vincenzo Librandi, Jun 12 2012 *)

G.f.: (1+2*z^2)/(1-z-2*z^3). - Simon Plouffe in his 1992 dissertation
a(n) = A077949(n+1) = |A077974(n+1)|.
a(n) = u(n+1) - 3*u(n) + 2*u(n-1) where u(i) = A003230(i) [Daykin and Tucker]. - N. J. A. Sloane, Jul 08 2014
a(n) = hypergeom([-n/3,-(1+n)/3,(1-n)/3],[-n/2,-(1+n)/2],-27/2) for n>=3. - Peter Luschny, Mar 09 2015

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

Original entry on oeis.org

1, 6, 9, 5, 6, 2, 0, 7, 6, 9, 5, 5, 9, 8, 6, 2, 0, 5, 7, 4, 1, 6, 3, 6, 7, 1, 0, 0, 1, 1, 7, 5, 3, 5, 3, 4, 2, 6, 1, 8, 1, 7, 9, 3, 8, 8, 2, 0, 8, 5, 0, 7, 7, 3, 0, 2, 2, 1, 8, 7, 0, 7, 2, 8, 4, 4, 5, 2, 4, 4, 5, 3, 4, 5, 4, 0, 8, 0, 0, 7, 2, 2, 1, 3, 9, 9

Offset: 1

Clark Kimberling, Jul 14 2017

			1.6956207695598620574163671001175353426181793882085077...

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.

Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
Index entries for algebraic numbers, degree 3.

Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.

Cf. A078140 (includes guide to constants similar to A289260).

z = 2000; r = 8/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289260 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289265 *)

solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019

r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019

A366525 Irregular triangular array read by rows: T(n,k) = number of partitions p of n such that f(p) = k >= 0, where f is defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 3, 3, 6, 2, 6, 6, 3, 5, 9, 5, 3, 7, 9, 9, 4, 1, 7, 13, 12, 6, 4, 10, 12, 15, 12, 5, 2, 7, 16, 19, 16, 12, 5, 2, 12, 16, 24, 22, 18, 6, 3, 11, 20, 28, 29, 24, 14, 6, 3, 12, 19, 31, 34, 36, 24, 13, 4, 3, 12, 23, 36, 42, 50, 30, 25, 8, 4

Offset: 1

Clark Kimberling, Oct 12 2023

For a partition p = (p(1),p(2),...,p(k)) of n, where p(1) >= p(2) >= ... >= p(k), define r(p) by subtracting 1 from p(1) and adding 1 to p(k) and then arranging the result in nonincreasing order. Iterating r eventually results in repetition; the function f(p) is the number of iterations of r up to but not including the first repeat. For example, starting with (5,3,2,2,1,1,1,1), the r-iterates are (4,3,2,2,2,1,1,1), (3,3,2,2,2,2,1,1), (3,2,2,2,2,2,2,1), (2,2,2,2,2,2,2,2), (3,2,2,2,2,2,2,1), so that f(5,3,2,2,1,1,1,1) = 3.

			First 18 rows:
   1
   1     1
   2     1
   2     3
   4     3
   3     6     2
   6     6     3
   5     9     5     3
   7     9     9     4     1
   7    13    12     6     4
  10    12    15    12     5     2
   7    16    19    16    12     5    2
  12    16    24    22    18     6    3
  11    20    28    29    24    14    6      3
  12    19    31    34    36    24    13     4     3
  12    23    36    42    50    30    25     8     4    1
  16    23    42    54    59    45    34    15     5    4
  13    28    47    57    74    61    52    28    16    5    4
Row 6 represents 3 partitions p that are self-repeating (i.e., k = 0), 6 such that f(p) = 1, and 2 such that f(p) = 2. Specifically,
  f(p) = 0 for these partitions: [6], [2,2,1,1], [2,1,1,1].
f(p) = 1 for these: [4,2], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [1,1,1,1,1,1].
f(p) = 2 for these: [5,1], [4,1,1].

John Tyler Rascoe, Rows n = 1..60, flattened

Cf. A000041 (row sums), A003479 (row lengths, after 2nd term).

Cf. A062968 (1st column).

r[list_] := If[Length[list] == 1, list, Reverse[Sort[# +
Join[{-1}, ConstantArray[0, Length[#] - 2], {1}]] &[Reverse[Sort[list]]]]];
f[list_] := NestWhileList[r, Reverse[Sort[list]], Unequal, All];
t = Table[BinCounts[#, {0, Max[#] + 1, 1}] &[Map[-1 + Length[Union[#]] &[f[#]] &, IntegerPartitions[n]]], {n, 1,20}]
Map[Length, t]; t1 = Take[t, 18]; TableForm[t1]
Flatten[t1]
(* Peter J. C. Moses, Oct 10 2023 *)

from sympy .utilities.iterables import ordered_partitions
from collections import Counter
def A366525_rowlist(row_n):
    A = []
    for i in range(1,row_n+1):
        A.append([]); p,C = list(ordered_partitions(i)),Counter()
        for j in range(0,len(p)):
            x,a1,a,b = 0,[],list(p[j]),list(p[j])
            while i:
                b[-1] -= 1; b[0] += 1
                if b[-1] == 0: b.pop(-1)
                b = sorted(b); x += 1
                if a == b or a1 == b:
                    C.update({x}); break
                else:
                    a1 = a.copy(); a = b.copy()
        for z in range(1,len(C)+1): A[i-1].append(C[z])
    return(A) # John Tyler Rascoe, Nov 09 2023

cp's OEIS Frontend

A003229 a(n) = a(n-1) + 2*a(n-3) with a(0)=a(1)=1, a(2)=3.

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A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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A366525 Irregular triangular array read by rows: T(n,k) = number of partitions p of n such that f(p) = k >= 0, where f is defined in Comments.

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