A049456 -id:A049456 - cp's OEIS frontend

A064881 Eisenstein array Ei(1,2).

1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 5, 4, 7, 3, 8, 5, 7, 2, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, 7, 16, 9, 11, 2

Offset: 1

Wolfdieter Lang, Oct 19 2001

In Eisenstein's notation this is the array for m=1 and n=2; see example in given reference p. 42. The array for m=n=1 is A049456.
For n >= 1, the number of entries of row n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 3*A007051(n-1).
The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the subtree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the reference see A002487 and the link) with root 1/2. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.

			{1,2};
{1,3,2};
{1,4,3,5,2};
{1,5,4,7,3,8,5,7,2}; ...
This binary subtree of rationals is built from
1/2;
1/3, 3/2;
1/4, 4/3, 3/5, 5/2; ...

R. Backhouse, J. F. Ferreira, On Euclid’s algorithm and elementary number theory, Sci. Comput. Program. 76, No. 3, 160-180 (2011).
N. Calkin and H. S. Wilf, Recounting the Rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850) 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
Index entries for sequences related to Stern's sequences

nmax = 6; a[n_, m_?EvenQ] := a[n - 1, m/2]; a[n_, m_?OddQ] := a[n, m] = a[n - 1, (m - 1)/2] + a[n - 1, (m + 1)/2]; a[1, 0] = 1; a[1, 1] = 2; Flatten[ Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* Jean-François Alcover, Sep 27 2011 *)
eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
Flatten@NestList[eisen, {1, 2}, 6] (* Harlan J. Brothers, Feb 18 2015 *)

a(n, m) = a(n-1, m/2) if m is even, else a(n, m) = a(n-1, (m-1)/2) + a(n-1, (m+1)/2), a(1, 0)=1, a(1, 1)=2.

A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 13, 13, 34, 34, 89, 89, 233, 233, 610, 610, 1597, 1597, 4181, 4181, 10946, 10946, 28657, 28657, 75025, 75025, 196418, 196418, 514229, 514229, 1346269, 1346269, 3524578, 3524578, 9227465, 9227465, 24157817, 24157817, 63245986, 63245986, 165580141, 165580141

Offset: 0

Paul Barry, May 26 2004

Fibonacci(2*n+1) repeated. a(n) is the right neighbor of Fibonacci(n+2) in A049456 and A002487 (starts 2,2,5,...). A000045(n+2) = A094966(n) + a(n).
Diagonal sums of A109223. - Paul Barry, Jun 22 2005
The Fi2 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 11 2011
a(n) is the last term of (n+1)-th row in Wythoff array A003603. -Reinhard Zumkeller, Jan 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
Ying Wang and Zihao Zhang, Hankel Determinants for a Class of Weighted Lattice Paths, arXiv:2409.18609 [math.CO], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A001519, A133080.

List([0..50], n -> Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2); # G. C. Greubel, Nov 18 2018

[IsEven(n) select Fibonacci(n+1) else Fibonacci(n): n in [0..70]]; // Vincenzo Librandi, Nov 18 2018

A094967 := proc(n) combinat[fibonacci](2*floor(n/2)+1) ; end proc: seq(A094967(n), n=0..37);

LinearRecurrence[{0,3,0,-1},{1,1,2,2},40] (* Harvey P. Dale, Apr 05 2015 *)
f[n_]:=If[OddQ@n, (Fibonacci[n]), Fibonacci[n+1]]; Array[f, 100, 0] (* Vincenzo Librandi, Nov 18 2018 *)
Table[Fibonacci[n, 0]*Fibonacci[n] + LucasL[n, 0]*Fibonacci[n + 1]/2, {n, 0, 50}] (* G. C. Greubel, Nov 18 2018 *)

vector(50, n, n--; fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2) \\ G. C. Greubel, Nov 18 2018

[fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2 for n in range(50)] # G. C. Greubel, Nov 18 2018

G.f.: (1+x-x^2-x^3)/(1-3*x^2+x^4).
a(n) = Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2)+k, 2*k). - Paul Barry, Jun 22 2005
Starting (1, 2, 2, 5, 5, 13, 13, ...) = A133080 * A000045, where A000045 starts with "1". - Gary W. Adamson, Sep 08 2007
a(n) = Fibonacci(n+1)^(4*k+3) mod Fibonacci(n+2), for any k>-1, n>0. - Gary Detlefs, Nov 29 2010

A355855 A family of triangles T(m), m > 0, read by triangles and then by rows; triangle T(1) is [1; 1, 1]; for m > 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; t+u, t+v; u, u+v, v].

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 2, 4, 2, 3, 4, 4, 3, 1, 3, 2, 3, 1, 1, 4, 4, 3, 6, 3, 5, 7, 7, 5, 2, 6, 4, 6, 2, 5, 6, 8, 8, 6, 5, 3, 7, 4, 8, 4, 7, 3, 4, 6, 7, 6, 6, 7, 6, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 1, 5, 5, 4, 8, 4, 7, 10, 10, 7, 3, 9, 6, 9, 3, 8, 10, 13, 13, 10, 8

Offset: 1

Rémy Sigrist, Jul 19 2022

We apply the following substitutions to transform T(m) into T(m+1):
                           t
                          / \
        t                /   \
       / \     -->     t+u---t+v
      u---v            / \   / \
                      /   \ /   \
                     u----u+v----v
This sequence can be seen as a two-dimensional variant of A049456.
The base of T(m) corresponds to the m-th row of A049456.
T(m) has 2^(m-1)+1 rows, and largest term 2^(m-1).
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

			T(1) is:
          1
         1 1
T(2) is:
          1
         2 2
        1 2 1
T(3) is:
          1
         3 3
        2 4 2
       3 4 4 3
      1 3 2 3 1
T(4) is:
          1
         4 4
        3 6 3
       5 7 7 5
      2 6 4 6 2
     5 6 8 8 6 5
    3 7 4 8 4 7 3
   4 6 7 6 6 7 6 4
  1 4 3 5 2 5 3 4 1

Rémy Sigrist, Table of n, a(n) for n = 1..11313
Rémy Sigrist, Representation of multiples of 2 in T(12)
Rémy Sigrist, Representation of multiples of 2^2 in T(12)
Rémy Sigrist, Representation of multiples of 3 in T(12)
Rémy Sigrist, Representation of multiples of 5 in T(12)
Rémy Sigrist, Colored representation of T(10) (the color is function of T(10)(n,k))
Rémy Sigrist, PARI program
Rémy Sigrist, Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp, arXiv:2301.06039 [math.CO], 2023.

Cf. A049456.

PARI
```
See Links section.
```

T(m,n,k) = { if (m==1, 1, my (nn=(n+1)\2, kk=(k+1)\2); if (n%2==1 && k%2==1, T(m-1, nn, kk), n%2==1 && k%2==0, T(m-1, nn, kk) + T(m-1, nn, kk+1), n%2==0 && k%2==1, T(m-1, nn, kk) + T(m-1, nn+1, kk), T(m-1, nn, kk) + T(m-1, nn+1, kk+1))) }

A070878 Stern's diatomic array read by rows (version 2).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5

Offset: 0

N. J. A. Sloane, May 20 2002

Row n has length 2^n + 1.

			Triangle begins:
1,0;
1,1,0;
1,2,1,1,0;
1,3,2,3,1,2,1,1,0;
...

Harvey P. Dale, Table of n, a(n) for n = 0..10000
C. Giuli and R. Giuli, A primer on Stern's diatomic sequence, Fib. Quart., 17 (1979), 103-108, 246-248 and 318-320 (but beware errors).
Index entries for sequences related to Stern's sequences

Cf. A049456, A070879, A049455.

Rows sums are A007051. See A293160 for number of distinct terms in each row.

row[1] = {1, 0}; row[n_] := row[n] = (r = row[n-1]; Riffle[r, Most[r + RotateLeft[r]]]); Flatten[ Table[row[n], {n, 1, 7}]] (* Jean-François Alcover, Nov 03 2011 *)
Flatten[NestList[Riffle[#,Total/@Partition[#,2,1]]&,{1,0},6]] (* Harvey P. Dale, Dec 06 2014 *)

Each row is obtained by copying the previous row but interpolating the sums of pairs of adjacent terms. E.g. after 1 2 1 1 0 we get 1 1+2 2 2+1 1 1+1 1 1+0 0.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 07 2003

A070879 Stern's diatomic array read by rows (version 3 - same as version 2, A070878, but with final 0 in each row omitted).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 2, 1, 1, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 5, 4, 7, 3, 8

Offset: 0

N. J. A. Sloane, May 20 2002

Row n has length 2^n.
From Yosu Yurramendi, Apr 08 2019: (Start)
The terms (n>0) may be written as a left-justified array with rows of length 2^m:
1,
1, 1,
1, 2, 1, 1,
1, 3, 2, 3, 1, 2, 1, 1,
1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1,
1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4,...
as well as right-justified fashion:
                                                                                1,
                                                                             1, 1,
                                                                       1, 2, 1, 1,
                                                           1, 3, 2, 3, 1, 2, 1, 1,
                                   1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1,
     ... , 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1,
...
For properties see FORMULA section.
(End)

C. Giuli and R. Giuli, A primer on Stern's diatomic sequence, Fib. Quart., 17 (1979), 103-108, 246-248 and 318-320 (but beware errors).
Index entries for sequences related to Stern's sequences

Cf. A049456, A070878, A049455.

Rows sums are A007051.

From Yosu Yurramendi, Apr 08 2019: (Start)
a(2^(m+1)+k-1) = A002487(2^m+k); a(2^(m+1)+2^m+k-1) = a(2^m+k-1)  for m >= 0, 0 <= k < 2^m.
a(2^(m+1)-1-(k+1)) = A002487(k+1); a(2^(m+1)+k) - a(2^m+k) = A002487(k)   for m >= 0, 0 <= k < 2^m.
a(2^m-1) = 1 for m >= 0; a(2^(m+1)+k-1) = a(2^(m+1)-k-1) + a(2^m+k-1) for m >= 0, 0 < k < 2^m.
a(2^m+2^m'+k'-1) = a(2^(m'+1)+k'-1)*(m-m'-1) + a(2^m'+k'-1) for m >= 1, 0 <= m' < m, 0 <= k' < 2^m'.
(End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 07 2003

A094966 Left-hand neighbors of Fibonacci numbers in Stern's diatomic series.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 8, 21, 21, 55, 55, 144, 144, 377, 377, 987, 987, 2584, 2584, 6765, 6765, 17711, 17711, 46368, 46368, 121393, 121393, 317811, 317811, 832040, 832040, 2178309, 2178309, 5702887, 5702887, 14930352, 14930352, 39088169, 39088169

Offset: 0

Paul Barry, May 26 2004

Fibonacci(2n) repeated. a(n) is the left neighbor of Fibonacci(n+2) in A002487 and A049456. A000045(n+2) = a(n)+A094967(n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A001906.

[Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Mar 29 2016

CoefficientList[Series[x (1 + x)/(1 - 3 x^2 + x^4), {x, 0, 38}], x] (* Michael De Vlieger, Mar 28 2016 *)

concat(0, Vec(x*(1+x)/(1-3*x^2+x^4) + O(x^50))) \\ Colin Barker, Mar 28 2016

G.f.: x*(1+x) / (1-3*x^2+x^4).
a(n) = Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2.
a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-3+sqrt(5)) - (-1-sqrt(5))^n*(-1+sqrt(5)) - (-1+sqrt(5))^n - sqrt(5)*(-1+sqrt(5))^n + 3*(1+sqrt(5))^n + sqrt(5)*(1+sqrt(5))^n))/sqrt(5). - Colin Barker, Mar 28 2016

A337277 Stern's triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1

Offset: 0

N. J. A. Sloane, Sep 09 2020

The first two rows are 1, then 1,1,1. To get row n, copy row n-1, and insert c+d between every pair of adjacent terms c,d, and finally insert a 1 at the beginning and end of the row.

The maximum value in row n is A000045(n+1). - Alois P. Heinz, Sep 09 2020

			Triangle begins:
  1;
  1, 1, 1;
  1, 1, 2, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1;
  ...

Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Alois P. Heinz, Rows n = 0..14, flattened
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019.

Cf. A000045, A002487, A049456, A334627.
Row sums give A000244.
Row lengths give A126646.

T:= proc(n) option remember; `if`(n=0, 1, (L-> [1, L[1], seq(
      [L[i-1]+L[i], L[i]][], i=2..nops(L)), 1][])([T(n-1)]))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 09 2020

Nest[Append[#, Flatten@ Join[{1}, If[Length@ # > 1, Map[{#1, #1 + #2} & @@ # &, Partition[#[[-1]], 2, 1] ], {}], {#[[-1, -1]]}, {1}]] &, {{1}}, 5] // Flatten (* Michael De Vlieger, Sep 09 2020 *)

T(n,n) = A002487(n+1). - Alois P. Heinz, Sep 09 2020

A064886 Eisenstein array Ei(2,3).

Original entry on oeis.org

2, 3, 2, 5, 3, 2, 7, 5, 8, 3, 2, 9, 7, 12, 5, 13, 8, 11, 3, 2, 11, 9, 16, 7, 19, 12, 17, 5, 18, 13, 21, 8, 19, 11, 14, 3, 2, 13, 11, 20, 9, 25, 16, 23, 7, 26, 19, 31, 12, 29, 17, 22, 5, 23, 18, 31, 13, 34, 21, 29, 8, 27, 19, 30

Offset: 1

Wolfdieter Lang, Oct 19 2001

In Eisenstein's notation this is the array for m=2 and n=3; see pp. 41-2 of the Eisenstein reference given for A064881. This is identical with the array for m=3, n=2, given in A064885, read backwards. The array for m=n=1 is A049456.
For n >= 1, the number of entries of row n >= 1 is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 5*A007051(n-1).
The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the subtree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the reference see A002487, also for the Wilf link) with root 2/3. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.

			{2,3}; {2,5,3}; {2,7,5,8,3}; {2,9,7,12,5,13,8,11,3}; ...
This binary subtree of rationals is built from 2/3; 2/5,5/3; 2/7,7/5,5/8,8/3; ...

Index entries for sequences related to Stern's sequences

a[1, 0] = 2; a[1, 1] = 3; a[n_ /; n >= 1, m_ /; m >= 0] := If[EvenQ[m], a[n, m] = a[n-1, m/2], a[n, m] = a[n-1, (m-1)/2] + a[n-1, (m+1)/2]]; Table[a[n, m], {n, 1, 6}, {m, 0, 2^(n-1)}] // Flatten (* Jean-François Alcover, Feb 27 2018 *)

a(n, m)= a(n-1, m/2) if m is even, else a(n, m)= a(n-1, (m-1)/2)+a(n-1, (m+1)/2), a(1, 0)=2, a(1, 1)=3.

A174868 Partial sums of Stern's diatomic series A002487.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset: 0

Jonathan Vos Post, Dec 01 2010

After the initial 0, identical to A007729.

			a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1)),initial=0)
A174868_list = list(islice(A174868_gen(),30)) # Chai Wah Wu, May 07 2023

a(n) = Sum_{i=0..n} A002487(i).
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017
a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018
a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

A293165 Number of distinct terms in row n of A049455.

Original entry on oeis.org

2, 2, 3, 4, 6, 8, 14, 21, 32, 49, 79, 119, 192, 301, 466, 735, 1176, 1851, 2927, 4598, 7297, 11553, 18279, 28864, 45833, 72357, 114743, 181722, 287927, 455749, 722459, 1144371, 1813976, 2873752, 4553644, 7213621, 11432170, 18120734, 28716295, 45491134

Offset: 1

N. J. A. Sloane, Oct 15 2017

nonn

Equals A293160(n+1)+1.

Cf.A049455, A049456, A293160.

cp's OEIS Frontend

A064881 Eisenstein array Ei(1,2).

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A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.

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A355855 A family of triangles T(m), m > 0, read by triangles and then by rows; triangle T(1) is [1; 1, 1]; for m > 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; t+u, t+v; u, u+v, v].

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PARI

PARI

A070878 Stern's diatomic array read by rows (version 2).

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Mathematica

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A070879 Stern's diatomic array read by rows (version 3 - same as version 2, A070878, but with final 0 in each row omitted).

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A094966 Left-hand neighbors of Fibonacci numbers in Stern's diatomic series.

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Magma

Mathematica

PARI

Formula

A337277 Stern's triangle read by rows.

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