A105163 -id:A105163 - cp's OEIS frontend

A270809 a(n) = n^3/3 - 7*n/3 + 4.

4, 2, 2, 6, 16, 34, 62, 102, 156, 226, 314, 422, 552, 706, 886, 1094, 1332, 1602, 1906, 2246, 2624, 3042, 3502, 4006, 4556, 5154, 5802, 6502, 7256, 8066, 8934, 9862, 10852, 11906, 13026, 14214, 15472, 16802, 18206, 19686, 21244, 22882, 24602, 26406, 28296, 30274

Offset: 0

N. J. A. Sloane, Apr 06 2016

M. Diepenbroek, M. Maus, A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Theorem 3.14.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A105163.

[n^3/3-7*n/3+4: n in [0..50]]; // Vincenzo Librandi, Apr 08 2016

Table[n^3 / 3 - 7 n / 3 + 4, {n, 0, 50}] (* Vincenzo Librandi, Apr 08 2016 *)
LinearRecurrence[{4,-6,4,-1},{4,2,2,6},50] (* Harvey P. Dale, Jul 18 2025 *)

vector(50, n, n--; n^3/3-7*n/3+4) \\ Bruno Berselli, Apr 08 2016

x='x+O('x^99); Vec((4-14*x+18*x^2-6*x^3)/(1-x)^4) \\ Altug Alkan, Apr 08 2016

[n^3/3-7*n/3+4 for n in [0..50]] # Bruno Berselli, Apr 08 2016

O.g.f.: (4 - 14*x + 18*x^2 - 6*x^3)/(1-x)^4. - Vincenzo Librandi, Apr 08 2016
E.g.f.: (12 - 6*x + 3*x^2 + x^3)*exp(x)/3. - Bruno Berselli, Apr 08 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Apr 08 2016
a(n) = 2*A105163(n) for n>0. - Bruno Berselli, Apr 08 2016

A363256 Number of length n strings on the alphabet {0,1,2,3} with digit sum at most 4.

Original entry on oeis.org

1, 4, 13, 32, 66, 121, 204, 323, 487, 706, 991, 1354, 1808, 2367, 3046, 3861, 4829, 5968, 7297, 8836, 10606, 12629, 14928, 17527, 20451, 23726, 27379, 31438, 35932, 40891, 46346, 52329, 58873, 66012, 73781, 82216, 91354, 101233, 111892, 123371, 135711

Offset: 0

Daniel T. Martin, May 23 2023

			For n=2, the 13 strings are all possible 2-character strings of '0', '1', '2' and '3' except the four strings '33', '32', '23'.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A227259 (the same for {0,1,2} with digit sum <= 4).
Cf. A105163 (the same for {0,1,2} with digit sum <= 3, shifted by 2).
Cf. A005718.

f[n_, r_, l_] := If[r < 0, 0, If[r==0, 1, If[l < 0, 0, If[l == 0, 1, Sum[f[n, r-j, l-1], {j, 0, n}]]]]]; Table[f[3, 4,x], {x, 0, 40}]

a(n) = (((n + 10)*n + 35)*n + 26)*n/24 + 1.
G.f.: -(x^4 - 3*x^3 + 3*x^2 - x + 1)/(x - 1)^5.
a(n) = 1 + A005718(n-1) for n>=1.

cp's OEIS Frontend

A270809 a(n) = n^3/3 - 7*n/3 + 4.

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