A046698 -id:A046698 - cp's OEIS frontend

A085165 A057163-conjugate of A085159.

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 12, 16, 17, 21, 10, 18, 11, 14, 15, 13, 19, 20, 22, 23, 26, 30, 31, 35, 42, 43, 44, 45, 49, 56, 57, 58, 63, 24, 32, 47, 46, 59, 25, 48, 28, 37, 38, 29, 39, 40, 41, 27, 50, 33, 51, 52, 34, 53, 54, 55, 36, 60, 61, 62, 64, 65, 68, 72, 73, 77

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085166. a(n) = A057163(A085159(A057163(n))) = A085162(A085166(A085162(n))). Occurs in A073200. Cf. also A085162, A086429, A086430.

Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082314 Involution of natural numbers: A057502-conjugate of A057164.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 11, 10, 12, 13, 21, 22, 20, 17, 18, 19, 16, 14, 15, 23, 28, 25, 30, 33, 24, 29, 26, 31, 34, 27, 32, 35, 36, 58, 62, 59, 63, 64, 57, 61, 54, 45, 48, 55, 46, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 39, 52, 43, 38, 40, 41, 65, 79, 70, 84, 93

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A057502(A069889(n)). Occurs in A073200 as row 2361759710983228099211. Cf. also A082313.

Number of cycles: A007123. Number of fixed-points: A001405. Max. cycle size: A046698. LCM of cycle sizes: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

a(n) = A057502(A057164(A057501(n)))

A085166 A057163-conjugate of A085160.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 10, 19, 17, 18, 11, 12, 15, 20, 21, 13, 22, 23, 37, 42, 24, 51, 44, 47, 25, 26, 38, 53, 56, 27, 60, 45, 46, 48, 49, 50, 28, 29, 30, 31, 40, 39, 43, 32, 52, 54, 55, 57, 58, 59, 33, 34, 35, 41, 61, 62, 63, 36, 64, 65, 107, 121, 66, 149

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085165. a(n) = A057163(A085160(A057163(n))) = A085162(A085165(A085162(n))). Occurs in A073200. Cf. also A085162, A086429, A086430.

Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A129687 A129686 * A007318.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 2, 6, 7, 4, 1, 2, 8, 13, 11, 5, 1, 2, 10, 21, 24, 16, 6, 1, 2, 12, 31, 45, 40, 22, 7, 1, 2, 14, 43, 76, 85, 62, 29, 8, 1, 2, 16, 57, 119, 161, 147, 91, 37, 9, 1, 2, 18, 73, 176, 280, 308, 238, 128, 46, 10, 1, 2, 20, 91, 249, 456

Offset: 0

Gary W. Adamson, Apr 28 2007

Row sums = A084215: (1, 2, 5, 10, 20, 40, 80, ...). A007318 * A129686 = A124725.
From Philippe Deléham, Feb 12 2014: (Start)
Riordan array ((1+x^2)/(1-x), x/(1-x)).
Diagonal sums are A000032(n) - 0^n (cf. A000204).
T(n,0) = A046698(n+1).
T(n+1,1) = A004277(n).
T(n+2,2) = A002061(n+1).
T(n+3,3) = A006527(n+1) = A167875(n).
T(n+4,4) = A006007(n+1).
T(n+5,5) = A081282(n+1). (End)

			First few rows of the triangle:
  1;
  1,   1;
  2,   2,   1;
  2,   4,   3,   1;
  2,   6,   7,   4,   1;
  2,   8,  13,  11,   5,   1;
  2,  10,  21,  24,  16,   6,   1;
  2,  12,  31,  45,  40,  22,   7,   1;
  2,  14,  43,  76,  85,  62,  29,   8,   1;
  2,  16,  57, 119, 161, 147,  91,  37,   9,   1;
  ...

Cf. A129686, A007318, A124725.

Cf. Columns: A046698, A004277, A002061, A006527, A006007, A081282

A129686 * A007318 (Pascal's Triangle), as infinite lower triangular matrices.

T(n,k) = T(n-1,k) + T(n-1,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

More terms from Philippe Deléham, Feb 12 2014

A287597 a(0) = 0, a(1) = 1; a(2n) = n - a(a(n)), a(2n+1) = a(a(n)) + a(a(n+1)).

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 1, 4, 1, 4, 1, 6, 1, 6, 3, 6, 3, 8, 3, 8, 3, 10, 3, 10, 3, 12, 3, 12, 3, 14, 3, 14, 3, 16, 5, 14, 5, 18, 5, 16, 5, 20, 5, 18, 5, 22, 5, 20, 5, 24, 7, 20, 7, 26, 7, 22, 7, 28, 7, 24, 7, 30, 7, 26, 7, 32, 7, 28, 7, 34, 7, 30, 7, 36, 9, 30, 9, 38, 7, 34, 7, 40, 9, 34, 9

Offset: 0

Ilya Gutkovskiy, May 27 2017

nonn

A variation on Hofstadter's G-sequence and Stern's diatomic sequence.

Eric Weisstein's World of Mathematics, Stern's Diatomic Series
Eric Weisstein's World of Mathematics, Hofstadter G-Sequence
Index entries for sequences related to Stern's sequences
Index entries for Hofstadter-type sequences

Cf. A002487, A005206, A046698, A287475, A287476, A287477.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], n/2 - a[a[n/2]], a[a[(n - 1)/2]] + a[a[(n + 1)/2]]]; Table[a[n], {n, 0, 85}]

a(2*n) + a(2*n+1) + a(2*n+2) = 2*n + 1 (from definition).

A308818 a(n) = a(a(n-1) mod n) + a(a(n-2) mod n) with a(0)=2 and a(1)=3.

Original entry on oeis.org

2, 3, 5, 7, 10, 7, 13, 15, 22, 23, 12, 6, 15, 18, 13, 25, 41, 37, 10, 22, 17, 40, 47, 40, 81, 38, 22, 53, 85, 134, 51, 29, 156, 215, 23, 47, 46, 35, 69, 98, 144, 81, 108, 116, 102, 37, 47, 37, 72, 75, 85, 104, 217, 111, 10, 15, 37, 60, 40, 147, 197, 51, 110

Offset: 0

Arran Ireland, Jun 26 2019

nonn

a(0) and a(1) are chosen to be the smallest starting numbers greater than 1 that are believed to result in a sequence that doesn't cycle.
Empirical observation of the first 10^8 terms suggests that the sequence doesn't enter a cycle.
Conjectures: (i) This sequence doesn't enter a cycle. (ii) There is an integer greater than 1 that can never appear in this sequence.

			a(2) = a(a(2-1) mod 2) + a(a(2-2) mod 2) = a(a(1) mod 2) + a(a(0) mod 2) = a(3 mod 2) + a(2 mod 2) = a(1) + a(0) = 3 + 2 = 5.

Cf. A005185, A046698, A003160.

Cf. A000027 (if a(0)=1 and a(1)=2).

a = [2, 3]
for n in range(2, 10**4 + 3):
    a.append(a[(a[n - 1] % n)] + a[(a[n - 2] % n)])
    print((n - 2), ",", a[n - 2], sep="")

A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Philippe Deléham, Nov 02 2009

Diagonal sums: A060576.

A167374*A154325 formatted as lower triangular matrix. - Philippe Deléham, Nov 19 2009

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

Cf. A097806, A103451.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A046698(n+1), A111286(n+1), A027327(n) for x= 0, 1, 2, 3 respectively.
G.f.: (1+x^2*y)/(1-x*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k-1) for n > 2, T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A085165 A057163-conjugate of A085159.

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A082314 Involution of natural numbers: A057502-conjugate of A057164.

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A085166 A057163-conjugate of A085160.

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A129687 A129686 * A007318.

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A287597 a(0) = 0, a(1) = 1; a(2*n) = n - a(a(n)), a(2*n+1) = a(a(n)) + a(a(n+1)).

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Mathematica

Formula

A308818 a(n) = a(a(n-1) mod n) + a(a(n-2) mod n) with a(0)=2 and a(1)=3.

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Python

A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Formula

A287597 a(0) = 0, a(1) = 1; a(2n) = n - a(a(n)), a(2n+1) = a(a(n)) + a(a(n+1)).