Daniel T. Martin - cp's OEIS frontend

User: Daniel T. Martin

Daniel T. Martin has authored 2 sequences.

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

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.

A282087 Number of length-n ternary strings that do not contain both "00" and "11".

Original entry on oeis.org

1, 3, 9, 27, 79, 229, 657, 1871, 5295, 14909, 41801, 116783, 325287, 903741, 2505377, 6932479, 19151519, 52833853, 145578265, 400705135, 1101936119, 3027902045, 8314284721, 22816209855, 62579270191, 171559358493, 470132335209, 1287861941487, 3526800739399

Offset: 0

Daniel T. Martin, Feb 05 2017

			for n=5 the 229 acceptable ternary strings are all length 5 strings of '0', '1', and '2' _except_ '00011', '00110', '00111', '00112', '00211', '01100', '10011', '11000', '11001', '11002', '11100', '11200', '20011', '21100'.

Colin Barker, Table of n, a(n) for n = 0..1000
Math StackExchange, Find a recurrence relation for the number of ternary strings of length ? that do not contain two consecutive 0's and two consecutive 1's.
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6,-2).

Cf. A078057 (number of length-n ternary strings that contain neither "00" nor "11").

Table[3^n, {n, 0, 3}]~Join~LinearRecurrence[{4, -1, -6, -2}, {79, 229, 657, 1871}, 24] (* or *)
Table[Count[Tuples[Range[0, 2], n], w_ /; Boole[SequenceCount[w, {0, 0}] > 0] Boole[SequenceCount[w, {1, 1}] > 0] == 0], {n, 0, 12}] (* Michael De Vlieger, Feb 05 2017, latter program version 10.1 *)

a(n)=([0,1,0,0;0,0,1,0;0,0,0,1;-2,-6,-1,4]^n*[1;3;9;27])[1,1] \\ Charles R Greathouse IV, Feb 05 2017

Vec((1 + x)*(1 - 2*x) / ((1 - 2*x - x^2)*(1 - 2*x - 2*x^2)) + O(x^30)) \\ Colin Barker, Feb 07 2017

import itertools
# Not feasible on most machines for large numbers
def find_a_sub_n(n):
    c = 0
    for q in itertools.product(*([['0','1','2']]*n)):
        h = ''.join(q)
        if not (('11' in h) and ('00' in h)):
            c = c+1
    return c

a(n) = 4*a(n-1) - a(n-2) - 6*a(n-3) - 2*a(n-4) for n >= 4 (derived in the math.stackexchange.com link).
From Colin Barker, Feb 07 2017: (Start)
a(n) = -(1-u)^(1+n)/2 - (1+u)^(1+n)/2 + (1-v)^n - (2*(1-v)^n)/v + (1+v)^n + (2*(1+v)^n) / v where u=sqrt(2) and v=sqrt(3).
G.f.: (1 + x)*(1 - 2*x) / ((1 - 2*x - x^2)*(1 - 2*x - 2*x^2)).
(End)

cp's OEIS Frontend

User: Daniel T. Martin

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

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