A380651 -id:A380651 - cp's OEIS frontend

A380024 a(n) = 4^n - 3^n - binomial(n,2)*3^(n-2).

0, 1, 6, 28, 121, 511, 2152, 9094, 38563, 163729, 694282, 2934592, 12348541, 51697075, 215291356, 891989002, 3677964295, 15099277669, 61745907934, 251632677604, 1022414950465, 4143511249831, 16755357788176, 67628131638478, 272531374722091

Offset: 0

Enrique Navarrete, Feb 05 2025

a(n) is the number of words of length n defined on 4 letters where one of the letters is used at least once but not twice.

			For n=2, the 6 words on {0, 1, 2, 3} that use 0 at least once but not twice are 10, 01, 20, 02, 30, 03.

Index entries for linear recurrences with constant coefficients, signature (13,-63,135,-108).

Cf. A005061, A086443, A380651.

Array[4^#-3^#-Binomial[#,2]*3^(#-2)&,25,0] (* or *) LinearRecurrence[{13,-63,135,-108},{0,1,6,28},25] (* James C. McMahon, Feb 14 2025 *)

def A380024(n): return (1<<(n<<1))-((n*(n-1)>>1)+9)*3**(n-2) if n>1 else n # Chai Wah Wu, Feb 14 2025

E.g.f.: exp(3*x)*(exp(x)-(x^2)/2-1).

G.f.: x*(1 - 7*x + 13*x^2)/((1 - 3*x)^3*(1 - 4*x)). - Stefano Spezia, Mar 03 2025

A380249 a(n) = 4^n - binomial(n,2)*3^(n-2).

Original entry on oeis.org

1, 4, 15, 55, 202, 754, 2881, 11281, 45124, 183412, 753331, 3111739, 12879982, 53291398, 220074325, 906337909, 3721011016, 15228417832, 62133328423, 252794939071, 1025901734866, 4153971603034, 16786738847785, 67722274817305, 272813804258572

Offset: 0

Enrique Navarrete, Feb 06 2025

a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used any number of times except twice.

			For n=2, the 15 words on {0, 1, 2, 3} that do not use 0 exactly twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 10, 01, 20, 02, 30, 03.

Index entries for linear recurrences with constant coefficients, signature (13,-63,135,-108).

Cf. A086443, A380651, A380024.

LinearRecurrence[{13,-63,135,-108},{1,4,15,55},25] (* Stefano Spezia, Mar 03 2025 *)

def A380249(n): return (1<<(n<<1))-(3**(n-2)*n*(n-1)>>1 if n>1 else 0) # Chai Wah Wu, Mar 15 2025

E.g.f.: exp(3*x)*(exp(x)-(x^2)/2).

G.f.: (1 - 9*x + 26*x^2 - 23*x^3)/((1 - 3*x)^3*(1 - 4*x)). - Stefano Spezia, Mar 03 2025

cp's OEIS Frontend

A380024 a(n) = 4^n - 3^n - binomial(n,2)*3^(n-2).

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