A007123 -id:A007123 - cp's OEIS frontend

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

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)))

A078925 Triangle of T1(n,m) = number of bracelets (necklaces that can be turned over) with m white beads and (2n+1-m) black ones, for 1<=m<=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 7, 10, 1, 5, 10, 20, 26, 1, 6, 14, 35, 57, 76, 1, 7, 19, 56, 111, 185, 232, 1, 8, 24, 84, 196, 392, 600, 750, 1, 9, 30, 120, 324, 756, 1368, 2052, 2494, 1, 10, 37, 165, 507, 1353, 2829, 4950, 7105, 8524, 1, 11, 44, 220, 759, 2277, 5412, 10824

Offset: 1

Thomas Hartinger (hartinger_t(AT)web.de), Dec 15 2002

Left half of odd rows of table A052307 with left column deleted.

			1; 1, 2; 1, 3, 4; 1, 4, 7, 10; ...

Index entries for sequences related to bracelets

Cf. A052307 for full table, A073020 for even number of beads. Last term in each row gives A007123.

Table[ f[n, 2n + 1], {n, 11}] (* Robert G. Wilson v, Mar 29 2006 *)

A378939 Number of Schroeder paths of semilength n up to reversal.

Original entry on oeis.org

1, 2, 5, 15, 54, 216, 947, 4375, 21018, 103550, 520041, 2649391, 13655190, 71053780, 372727751, 1968880111, 10463765490, 55909445082, 300160457453, 1618364548591, 8759315367894, 47574840887024, 259215969470139, 1416461749625543, 7760734001872842, 42624971709868054

Offset: 0

Andrew Howroyd, Dec 19 2024

nonn

A Schroeder path of semilength n is a path from (0,0) to (2n,0) using only steps U = (1,1), H = (2,0) and D = (1,-1). This sequence considers a path and its reversal to be the same.

			The a(1)..a(3) paths are:
a(1) = 1: H, UD;
a(2) = 5: HH, UHD, UDUD, UUDD, HUD=UDH;
a(3) = 15: HHH, HUDH, UHHD, UDHUD, UDUDUD, UUHDD, UUDUDD, UUUDDD, HHUD=UDHH, HUHD=UHDH, HUDUD=UDUDH, UHDUD=UDUHD, HUUDD=UDUDH, UHUDD=UUDHD, UDUUDD=UUDDUD.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A006318, A110110, A007123 (similar for Dyck paths), A378941 (similar for Motzkin paths).

seq(n) = { my(A=O(x^(n+2))); Vec(( -2*x - sqrt(1 - 6*x + x^2 + A) + sqrt(1 - 6*x^2 + x^4 + A)*(1 + x)/(1 - 2*x - x^2) ) / (4*x)) }

a(n) = (A006318(n) + A110110(n))/2.

G.f.: ( -2*x - sqrt(1 - 6*x + x^2) + sqrt(1 - 6*x^2 + x^4)*(1 + x)/(1 - 2*x - x^2) ) / (4*x).

A039791 Sequence arising in search for Legendre sequences.

Original entry on oeis.org

1, 1, 2, 4, 6, 14, 66, 95, 280, 1464, 2694, 10452, 41410, 95640, 323396, 1770963, 5405026, 13269146, 73663402, 164107650, 582538732, 3811895344, 7457847082, 30712068524, 151938788640, 353218528324, 1738341231644, 7326366290632, 17280039555348, 63583110959728

Offset: 1

N. J. A. Sloane

nonn

Number of bit strings of length L = 2n+1 and Hamming weight n (or n+1, as generated by Fletcher et al.) up to chord equivalence (i.e., up to color and general linear permutation x -> Ax+b mod L for A on Z/LZ* and b on Z/LZ--essentially a multiplicative necklace of phi(L) additive necklaces of L black and white beads where L is odd and the colors are as balanced as possible). The same strings are counted up to bracelet equivalence (x -> +-x+b mod L) at A007123, up to necklace equivalence (x -> x+b mod L) at A000108, and in full (x -> x) at A001700. - Travis Scott, Nov 24 2022

			From _Travis Scott_, Nov 24 2022: (Start)
If we decompose by weight the classes of period 2n+1 counted at A002729, a(n) appears as the twin towers of that triangle.
                              a(n)
                             |   |
                            (1) (1)
                         1   1   1   1
                     1   1   1   1   1   1
                 1   1   1   2   2   1   1   1
             1   1   2   3   4   4   3   2   1   1
         1   1   1   2   4   6   6   4   2   1   1  1
      1  1   1   3   7  10  14  14  10   7   3   1  1  1
   1  1  3   7  18  34  54  66  66  54  34  18   7  3  1  1
1  1  1  3  11  25  49  75  95  95  75  49  25  11  3  1  1  1. (End)

Roderick J. Fletcher, Marc Gysin, and Jennifer Seberry, Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices, Australasian J. Combin. 23 (2001), 75-86.

Coincides with A002995 offset by -1 at the A005097-th terms.

Cf. A000108, A007123, A001700.

Module[{a,b,g,L,m,x,z,Z},Table[L=2n+1;Z=Sum[Sum[Product[g=L/GCD[L,(k-1)i+j];Subscript[x,#]^(1/#)&@If[k==1,g,m=MultiplicativeOrder[k,g];g/GCD[g,(k^m-1)/(k-1)]m],{i,L}]L/GCD[L,k-1],{j,GCD[L,k-1]}],{k,Select[Range@L,CoprimeQ[#,L]&]}]/L/EulerPhi@L/.Subscript[x,z_]->a^z+b^z;CoefficientList[Z,{a,b}][[n+1,n+2]],{n,30}]] (* Travis Scott, Nov 24 2022 *)

a(n) ~ C(2n+1, n)/(2n+1)/phi(2n+1)

Empirical: a(n) == 1 (mod 2) for 2n+1 of the form 2^k+1 but not of the form p^2, else == 0.

More terms from Travis Scott, Nov 24 2022

cp's OEIS Frontend

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

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A078925 Triangle of T1(n,m) = number of bracelets (necklaces that can be turned over) with m white beads and (2n+1-m) black ones, for 1<=m<=n.

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A378939 Number of Schroeder paths of semilength n up to reversal.

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A039791 Sequence arising in search for Legendre sequences.

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