A274496 -id:A274496 - cp's OEIS frontend

A274497 Sum of the degrees of asymmetry of all binary words of length n.

0, 0, 2, 4, 16, 32, 96, 192, 512, 1024, 2560, 5120, 12288, 24576, 57344, 114688, 262144, 524288, 1179648, 2359296, 5242880, 10485760, 23068672, 46137344, 100663296, 201326592, 436207616, 872415232, 1879048192, 3758096384, 8053063680

Offset: 0

Emeric Deutsch, Jul 27 2016

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

A sequence is palindromic if and only if its degree of asymmetry is 0.

			a(3) = 4 because the binary words 000, 001, 010, 100, 011, 101, 110, 111 have degrees of asymmetry 0, 1, 0, 1, 1, 0, 1, 0, respectively.

Index entries for linear recurrences with constant coefficients, signature (2,4,-8).

Cf. A004526, A134353, A274496, A274498, A274499.

a:= proc(n) options operator, arrow: (1/8)*(2*n-1+(-1)^n)*2^n end proc: seq(a(n), n = 0 .. 30);

LinearRecurrence[{2, 4, -8}, {0, 0, 2}, 31] (* Jean-François Alcover, Nov 16 2022 *)

a(n)=(2*n-1+(-1)^n)*2^n/8 \\ Charles R Greathouse IV, Jul 08 2024

a(n) = (1/8)*(2n - 1 + (-1)^n)*2^n.
a(n) = Sum_{k>=0} k*A274496(n,k).
From Alois P. Heinz, Jul 27 2016: (Start)
a(n) = 2^(n-1) * A004526(n) = 2^(n-1)*floor(n/2).
a(n) = 2 * A134353(n-2) for n>=2. (End)
From Chai Wah Wu, Dec 27 2018: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) for n > 2.
G.f.: 2*x^2/((2*x - 1)^2*(2*x + 1)). (End)

A274498 Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).

Original entry on oeis.org

1, 3, 3, 6, 9, 18, 9, 36, 36, 27, 108, 108, 27, 162, 324, 216, 81, 486, 972, 648, 81, 648, 1944, 2592, 1296, 243, 1944, 5832, 7776, 3888, 243, 2430, 9720, 19440, 19440, 7776, 729, 7290, 29160, 58320, 58320, 23328, 729, 8748, 43740, 116640, 174960, 139968, 46656

Offset: 0

Emeric Deutsch, Jul 27 2016

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
A sequence is palindromic if and only if its degree of asymmetry is 0.
Sum(kT(n,k),k>=0) = A274499(n).

			From _Andrew Howroyd_, Jan 10 2018: (Start)
Triangle begins:
   1;
   3;
   3,   6;
   9,  18;
   9,  36,   36;
  27, 108,  108;
  27, 162,  324,  216;
  81, 486,  972,  648;
  81, 648, 1944, 2592, 1296;
  ...
(End)
T(2,0) = 3 because we have 00, 11, and 22.
T(2,1) = 6 because we have 01, 02, 10, 12, 20, and 21.

S. Elizalde, E. Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7 corollary 2.2 at m=3 and dropping each 2nd row.

Cf. A274496, A274497, A274499.

T := proc(n,k) options operator, arrow: 2^k*3^ceil((1/2)*n)*binomial(floor((1/2)*n), k) end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

T[n_, k_] := 2^k 3^Ceiling[n/2] Binomial[Floor[n/2], k];
Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Jan 04 2021 *)

T(n,k) = 2^k*3^ceil(n/2)*binomial(floor(n/2),k);
for(n=0, 10, for(k=0, n\2, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Jan 10 2018

T(n,k) = 2^k*3^ceiling(n/2)*binomial(floor(n/2),k).
G.f.: G(t,z) = (1 + 3z)/(1 - 3(1 + 2t)z^2).
The row generating polynomials P[n] satisfy P[n] = 3(1 + 2t)P[n-2] (n>=2). Easy to see if we note that the ternary words of length n (n>=2) are 0w0, 0w1, 0w2, 1w0, 1w1, 1w2, 2w0, 2w1, 2w2, where w is a ternary word of length n - 2.

A274499 Sum of the degrees of asymmetry of all ternary words of length n.

Original entry on oeis.org

0, 0, 6, 18, 108, 324, 1458, 4374, 17496, 52488, 196830, 590490, 2125764, 6377292, 22320522, 66961566, 229582512, 688747536, 2324522934, 6973568802, 23245229340, 69735688020, 230127770466, 690383311398, 2259436291848, 6778308875544, 22029503845518, 66088511536554

Offset: 0

Emeric Deutsch, Jul 27 2016

A sequence is palindromic if and only if its degree of asymmetry is 0.

			a(2) = 6 because the ternary words 00, 01, 02, 10, 11, 12, 20, 21, 22 have degrees of asymmetry 0, 1, 1, 1, 0, 1, 1, 1, 0, respectively.

Index entries for linear recurrences with constant coefficients, signature (3,9,-27).

Cf. A274496, A274497, A274498.

a := proc (n) options operator, arrow: (1/6)*(2*n-1+(-1)^n)*3^n end proc: seq(a(n), n = 0 .. 30);

LinearRecurrence[{3, 9, -27}, {0, 0, 6}, 28] (* Jean-François Alcover, Sep 09 2024 *)

a(n) = (1/6)*(2n - 1 + (-1)^n)*3^n.
a(n) = Sum(k*A274498(n,k), k>=0).
From Chai Wah Wu, Dec 27 2018: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) - 27*a(n-3) for n > 2.
G.f.: 6*x^2/((3*x - 1)^2*(3*x + 1)). (End)

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A274497 Sum of the degrees of asymmetry of all binary words of length n.

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A274498 Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).

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Author

Keywords

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Examples

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Crossrefs

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Maple

Mathematica

PARI

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A274499 Sum of the degrees of asymmetry of all ternary words of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula