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A240437 Number of non-palindromic n-tuples of 5 distinct elements. 3

%I

%S 0,20,100,600,3000,15500,77500,390000,1950000,9762500,48812500,

%T 244125000,1220625000,6103437500,30517187500,152587500000,

%U 762937500000,3814695312500,19073476562500,95367421875000,476837109375000,2384185742187500,11920928710937500,59604644531250000,298023222656250000

%N Number of non-palindromic n-tuples of 5 distinct elements.

%F a(n) = 1/2 * 5^(n/2) * ((sqrt(5)-1) * (-1)^n - sqrt(5)-1) + 5^n.

%F a(n) = 5^n - 5^ceiling(n/2).

%F a(n) = A000351(n) - A056451(n).

%F G.f.: (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3). [corrected by _Peter Luschny_, May 13 2019]

%e For n=3 a(3)=100 solutions are:

%e {0,0,1}, {0,0,2}, {0,0,3}, {0,0,4}, {0,1,1}, {0,1,2}, {0,1,3}, {0,1,4},

%e {0,2,1}, {0,2,2}, {0,2,3}, {0,2,4}, {0,3,1}, {0,3,2}, {0,3,3}, {0,3,4},

%e {0,4,1}, {0,4,2}, {0,4,3}, {0,4,4}, {1,0,0}, {1,0,2}, {1,0,3}, {1,0,4},

%e {1,1,0}, {1,1,2}, {1,1,3}, {1,1,4}, {1,2,0}, {1,2,2}, {1,2,3}, {1,2,4},

%e {1,3,0}, {1,3,2}, {1,3,3}, {1,3,4}, {1,4,0}, {1,4,2}, {1,4,3}, {1,4,4},

%e {2,0,0}, {2,0,1}, {2,0,3}, {2,0,4}, {2,1,0}, {2,1,1}, {2,1,3}, {2,1,4},

%e {2,2,0}, {2,2,1}, {2,2,3}, {2,2,4}, {2,3,0}, {2,3,1}, {2,3,3}, {2,3,4},

%e {2,4,0}, {2,4,1}, {2,4,3}, {2,4,4}, {3,0,0}, {3,0,1}, {3,0,2}, {3,0,4},

%e {3,1,0}, {3,1,1}, {3,1,2}, {3,1,4}, {3,2,0}, {3,2,1}, {3,2,2}, {3,2,4},

%e {3,3,0}, {3,3,1}, {3,3,2}, {3,3,4}, {3,4,0}, {3,4,1}, {3,4,2}, {3,4,4},

%e {4,0,0}, {4,0,1}, {4,0,2}, {4,0,3}, {4,1,0}, {4,1,1}, {4,1,2}, {4,1,3},

%e {4,2,0}, {4,2,1}, {4,2,2}, {4,2,3}, {4,3,0}, {4,3,1}, {4,3,2}, {4,3,3},

%e {4,4,0}, {4,4,1}, {4,4,2}, {4,4,3}.

%p gf := (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3): ser := series(gf, x, 26):

%p seq(coeff(ser,x,n), n=1..25); # _Peter Luschny_, May 13 2019

%t Table[1/2 * 5^(n/2) * ((Sqrt[5]-1) * (-1)^n - Sqrt[5]-1) + 5^n, {n, 25}]

%o (PARI) concat([0], Vec( ( (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3) + O(x^30) ) ) ) \\ _Joerg Arndt_, Aug 18 2014

%Y Cf. A233411, A242026, A242278.

%K nonn,easy

%O 1,2

%A _Mikk Heidemaa_, Aug 17 2014

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Last modified March 1 02:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)