cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A384198 a(n) = 3^(n-3)*(binomial(n,3) + 3*binomial(n,2) + 9*n + 27).

Original entry on oeis.org

1, 4, 16, 64, 255, 1008, 3942, 15228, 58077, 218700, 813564, 2991816, 10884699, 39208536, 139946130, 495303012, 1739406393, 6064804692, 21006799848, 72318491280, 247561692471, 843026984064, 2856838685886, 9637472084364, 32374793163285, 108327417770268, 361133233980372
Offset: 0

Views

Author

Enrique Navarrete, May 21 2025

Keywords

Comments

a(n) is the number of words of length n defined on 4 letters where a chosen letter (for example, the first letter of the alphabet) is used at most three times.

Examples

			a(5) = 1008 since from the 1024 words defined on {0, 1, 2, 3} we subtract the 5 permutations of 00001, the 5 permutations of 00002, the 5 permutations of 00003, and 00000.

Links

Crossrefs

Cf. A382618.

Programs

  • Mathematica
    LinearRecurrence[{12, -54, 108, -81}, {1, 4, 16, 64}, 30] (* or *)
A384198[n_] := 3^(n - 3)*(Binomial[n, 3] + 3*Binomial[n, 2] + 9*n + 27);
Array[A384198, 30, 0] (* Paolo Xausa, Jun 30 2025 *)

Formula

E.g.f.: (1 + x + x^2/2 + x^3/6)*exp(3*x).
G.f.: (1 - 8*x + 22*x^2 - 20*x^3)/(1 - 3*x)^4. - Stefano Spezia, May 22 2025

A382640 a(n) = 90*binomial(n,6) + 90*binomial(n,5) + 54*binomial(n,4) + 24*binomial(n,3) + 9*binomial(n,2) + 3*n + 1.

Original entry on oeis.org

1, 4, 16, 61, 217, 706, 2074, 5461, 12961, 28072, 56236, 105469, 187081, 316486, 514102, 806341, 1226689, 1816876, 2628136, 3722557, 5174521, 7072234, 9519346, 12636661, 16563937, 21461776, 27513604, 34927741, 43939561, 54813742, 67846606, 83368549, 101746561
Offset: 0

Views

Author

Enrique Navarrete, Apr 01 2025

Keywords

Comments

a(n) is the number of words of length n defined on 4 symbols where three chosen symbols (say, the three largest ones) are used at most twice.

Examples

			a(3) = 61 since from the 64 words defined on {0, 1, 2, 3} we subtract the three words 111, 222, 333.

Links

Crossrefs

Cf. A382618.

Formula

a(n) = 37 - 94*(n+1) + (187/2)*(n+1)^2 - (373/8)*(n+1)^3 + (103/8)*(n+1)^4 - (15/8)*(n+1)^5 + (1/8)*(n+1)^6.
E.g.f.: (1 + x + x^2/2)^3*exp(x).
G.f.: (1 - 3*x + 9*x^2 - 2*x^3 + 21*x^4 + 27*x^5 + 37*x^6)/(1 - x)^7. - Stefano Spezia, Apr 01 2025
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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