A083318 -id:A083318 - cp's OEIS frontend

A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

0, 1, 4, 2, 18, 6, 12, 3, 96, 24, 48, 8, 72, 15, 16, 5, 600, 120, 240, 30, 360, 56, 60, 10, 480, 87, 88, 20, 90, 21, 22, 7, 4320, 720, 1440, 144, 2160, 270, 288, 36, 2880, 416, 420, 67, 432, 73, 66, 13, 3600, 567, 568, 107, 570, 109, 108, 26, 576, 111, 112, 27, 114, 28, 52, 9, 35280, 5040, 10080, 840, 15120, 1584, 1680, 168

Offset: 0

Antti Karttunen, May 05 2015

This sequence can be represented as a binary tree. Each left hand child is produced as A255411(n), and each right hand child as A256450(n), when parent contains n >= 1:
                                      0
                                      |
                   ...................1...................
                  4                                       2
       18......../ \........6                  12......../ \........3
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   96       24         48       8          72       15         16       5
600  120 240  30    360  56   60 10     480  87   88  20     90  21   22 7
etc.
Because all terms of A255411 are even it means that odd terms can occur only in odd positions (together with some even terms, for each one of which there is a separate infinite cycle), while terms in even positions are all even.
After its initial 1, A255567 seems to give all the terms like 2, 3, 12, ... where the left hand child of the right hand child is one more than the right hand child of the left hand child (as for 2: 16 = 15+1, as for 3: 22 = 21+1, as for 12: 88 = 87+1).

Inverse: A255565.
Cf. A000142, A001511, A001563, A083318, A255411, A256450, A257679.
Cf. also A255567 and arrays A257503, A257505.
Related or similar permutations: A273666, A273667.

a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
Other identities:
For all n >= 0, a(2^n) = A001563(n+1). [The leftmost branch of the binary tree is given by n*n!]
For all n >= 0, a(A083318(n)) = A000142(n+1). [And the next innermost vertices by (n+1)! This follows because A256450(n*n! - 1) = (n+1)! - 1.]
For all n >= 1, A257679(a(n)) = A001511(n).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A287831 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 10, 96, 924, 8892, 85572, 823500, 7924932, 76265388, 733938084, 7063035084, 67970944260, 654116708844, 6294876045156, 60578584659468, 582976518206148, 5610260171812140, 53990200655546148, 519573366930788172, 5000101506310370436, 48118353758378062956

Offset: 0

David Nacin, Jun 02 2017

In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

Index entries for linear recurrences with constant coefficients, signature (9,6).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

Mathematica
```
LinearRecurrence[{9, 6}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+6*a(n-2)

a(n) = 9*a(n-1) + 6*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 6*x^2).
a(n) = ((1 - 11/sqrt(105))/2)*((9 - sqrt(105))/2)^n + ((1 + 11/sqrt(105))/2)*((9 + sqrt(105))/2)^n.

A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2a(h).

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 5, 31, 11, 63, 23, 127, 6, 47, 255, 13, 14, 95, 4, 511, 27, 29, 30, 191, 9, 1023, 55, 59, 61, 383, 19, 2047, 111, 119, 123, 767, 39, 4095, 223, 239, 247, 1535, 79, 8191, 447, 479, 495, 3071, 10, 159, 16383, 895, 62, 959, 991, 6143, 21, 319, 32767, 1791, 22, 125, 1919, 1983, 126, 12287, 46, 43, 639, 65535, 254, 3583, 12

Offset: 0

Antti Karttunen, May 05 2015

Because all terms of A255411 are even it means that even terms can only occur in even positions (together with some odd terms, for each one of which there is a separate infinite cycle), while terms in odd positions are all odd.

Inverse: A255566.
Cf. A000079, A000142, A001511, A001563, A083318, A255411, A256450, A257679, A257680, A257682, A257685.
Cf. also arrays A257503, A257505.
Related or similar permutations: A273665, A273668.

a(0) = 0; for n >= 1: if A257680(n) = 0 [i.e., n is one of the terms of A255411], then a(n) = 2*a(A257685(n)), otherwise [when n is one of the terms of A256450], a(n) = 1 + 2*a(A273662(n)).
Other identities:
For all n >= 1, A001511(a(n)) = A257679(n).
For all n >= 1, a(A001563(n)) = A000079(n-1) = 2^(n-1).
For all n >= 1, a(A000142(n)) = A083318(n-1).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 6, 4, 1, 2, 5, 10, 10, 5, 1, 2, 6, 15, 20, 15, 6, 1, 2, 7, 21, 35, 35, 21, 7, 1, 2, 8, 28, 56, 70, 56, 28, 8, 1, 2, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 2, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Gary W. Adamson, Aug 28 2007

Add 1 to all but the top entry in the left column of the Pascal matrix. - R. J. Mathar, Jan 18 2013

			First few rows of the triangle are:
  1;
  2, 1;
  2, 2,  1;
  2, 3,  3,  1;
  2, 4,  6,  4, 1;
  2, 5, 10, 10, 5, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A007318, A083318 (row sums), A103451.

A132749:= func< n,k | k eq n select 1 else k eq 0 select 2 else Binomial(n,k) >;
[A132749(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_]:= If[k==n, 1, If[k==0, 2, Binomial[n, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A132749(n,k): return 1 if k==n else 2 if k==0 else binomial(n,k)
flatten([[A132749(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

T(n,k) = A103451(n,k) * A007318(n,k), an infinite lower triangular matrix.
From G. C. Greubel, Feb 16 2021: (Start)
T(n,k) = binomial(n, k) with T(n, 0) = 2 for n>0.
Sum_{k=0..n} T(n, k) = A083318(n) = 2^n + 1^n - 0^n. (End)

More terms added by G. C. Greubel, Feb 16 2021

A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 8, 5, 1, 1, 2, 6, 17, 6, 1, 1, 2, 5, 18, 31, 7, 1, 1, 2, 5, 14, 47, 51, 8, 1, 1, 2, 5, 13, 41, 107, 78, 9, 1, 1, 2, 5, 13, 35, 115, 218, 113, 10, 1, 1, 2, 5, 13, 34, 98, 296, 407, 157, 11, 1, 1, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12

Offset: 1

Clark Kimberling, Mar 29 2012

Column 1: 1,1,1,1,1,1,1,1,1...
Row sums: A083318 (1+2^n)
Alternating row sums:  A137470
Limiting row:  1,1,2,5,13,34,..., odd-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.
For a discussion and guide to related arrays, see A208510.

			First six rows:
1
1...2
1...1...3
1...1...3....4
1...1...2....8...5
1...1...2....6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Cf. A210872, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

For a discussion and guide to related arrays, see A208510.
u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 5, 1, 0, 1, 4, 9, 1, 0, 1, 3, 12, 14, 1, 0, 1, 3, 9, 29, 20, 1, 0, 1, 3, 8, 27, 60, 27, 1, 0, 1, 3, 8, 22, 74, 111, 35, 1, 0, 1, 3, 8, 21, 63, 181, 189, 44, 1, 0, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 0, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 0, 1

Offset: 1

Clark Kimberling, Mar 29 2012

Column 1: 1,0,0,0,0,0,0,0,0,...
Row sums: A000225 (-1+2^n)
Alternating row sums:  (-1)*A077973
Limiting row:  0,1,3,8,21,..., even-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A000096 and U(n,n-2)=A086274.
For a discussion and guide to related arrays, see A208510.

			First six rows:
1
0...1
0...2...1
0...1...5...1
0...1...4...9....1
0...1...3...12...14...1
First three polynomials u(n,x): 1, x, 2x + x^2.

Cf. A210873, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.

A256537 First differences of corner sequence A256536 associated with A151723.

Original entry on oeis.org

1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65, 9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129

Offset: 1

Omar E. Pol, Apr 02 2015

Number of cells turned ON at n-th stage in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.

For an animation see "The movie version" in Links section.

			Written as an irregular triangle in which the row lengths are the absolute values of the terms of A141531, the sequence begins:
  1;
  3;
  5;
  9, 9;
  9, 17, 25, 17;
  9, 17, 29, 37, 33, 41, 57, 33;
  9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65;
  9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129;
  ...
It appears that the right border gives A083318, whose representation in base 2 gives A000533.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000533, A083318, A141531, A151723, A151724, A169779, A170898, A256536.

a(1) = 1; a(2) = 3.
It appears that a(n) = 1 + (A151724(n) + A151724(n-1))/3, n >= 3.
It appears that a(n) = 1 + (A151723(n) - A151723(n-2))/3, n >= 3.
It appears that a(n) = 1 + 2*(A170898(n-2) + A170898(n-3)), n >= 3.
a(3) = 5.
It appears that a(n) = 1 + 2*(A169779(n-2) - A169779(n-4)), n >= 4.

A341235 a(n) is the greatest term in n-th row of A341231.

Original entry on oeis.org

1, 2, 4, 4, 14, 6, 8, 8, 28, 14, 28, 12, 28, 14, 16, 16, 62, 28, 52, 20, 62, 28, 56, 24, 62, 28, 44, 28, 52, 30, 32, 32, 122, 62, 100, 36, 110, 52, 104, 40, 122, 62, 124, 44, 118, 56, 112, 48, 122, 62, 84, 52, 112, 54, 88, 56, 110, 58, 76, 60, 100, 62, 64, 64

Offset: 1

Rémy Sigrist, Feb 07 2021

Records of a(n)/n appear to happen for n in A083318.

			For n = 10:
- the trajectory of 10 under A245471 is 10 -> 5 -> 14 -> 7 -> 8 -> 4 -> 2 -> 1,
- so a(10) = 14.

Cf. A025586, A083318, A245471, A340873, A341231.

a(n) = { my (m=n); while (n>1, m=max(m, n=if (n%2, bitxor(n, 2*n+1), n/2))); m }

a(n) >= n, equality implies that n equals 1 or is even.

a(n) < 4*n.

A083319 4^n+3^n-2^n.

Original entry on oeis.org

1, 5, 21, 83, 321, 1235, 4761, 18443, 71841, 281315, 1106601, 4369403, 17304561, 68694995, 273202041, 1088057963, 4337948481, 17308878275, 69106635081, 276039644123, 1102997363601, 4408504767155, 17623562909721, 70462878967883

Offset: 0

Paul Barry, Apr 25 2003

Binomial transform of A083318

Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A083320.

CoefficientList[Series[(1-4x+2x^2)/((1-2x)(1-3x)(1-4x)), {x,0,40}],x]  (* Harvey P. Dale, Mar 28 2011 *)

a(n)=4^n+3^n-2^n.
G.f. (1-4x+2x^2)/((1-2x)(1-3x)(1-4x)).
E.g.f. exp(4x)+exp(3x)-exp(2x)

A103462 A triangle with palindromic cubes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 2, 9, 10, 5, 1, 1, 2, 17, 28, 17, 6, 1, 1, 2, 33, 82, 65, 26, 7, 1, 1, 2, 65, 244, 257, 126, 37, 8, 1, 1, 2, 129, 730, 1025, 626, 217, 50, 9, 1, 1, 2, 257, 2188, 4097, 3126, 1297, 344, 65, 10, 1, 1, 2, 513, 6562, 16385, 15626, 7777

Offset: 0

Paul Barry, Feb 07 2005

			Rows start {1}, {1,1}, {1,2,1}, {1,2,3,1}, {1,2,5,4,1},..

P. De Geest, World!Of Numbers

Columns include A040000, A083318, A103457, A046231, A046233, A103458, A103459, A000533. Cubes of column k are palindromic to base k, k>3 (start with column 0). Row sums are A103480. Diagonal sums are A103481.

Number triangle T(n, k)=if(k<=n, k^(n-k)+1-0^(n-k), 0); Column k has g.f. x^k(1-kx^2)/((1-x)(1-kx)).

cp's OEIS Frontend

A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

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A287831 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.

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A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2*a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2*a(h).

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A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

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A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

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A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

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Mathematica

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A256537 First differences of corner sequence A256536 associated with A151723.

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A341235 a(n) is the greatest term in n-th row of A341231.

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A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2a(h).