login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A131935 a(n) is the number of Khalimsky-continuous functions with four-point codomain and an n-point range. 4
4, 7, 15, 31, 65, 136, 285, 597, 1251, 2621, 5492, 11507, 24111, 50519, 105853 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
FORMULA
Let c^i(n) be the number of Khalimsky-continuous functions f from [0,n-1]_Z to [0,3]_Z such that f(n-1)=i for i=0,1,2,3 and let a(n) be their sum. Then a(n) = a(n-1)+2a(n-2)+c^1(n-3)+c^2(n-3)
The sequence is determined by the above recurrence together with the following recurrences:
c^0(2k + 1) = c^0(2k) + c^1(2k),
c^1(2k + 1) = c^1(2k),
c^2(2k + 1) = c^1(2k) + c^2(2k) + c^3(2k),
c^3(2k + 1) = c^3(2k) and
c^0(2k) = c^0(2k - 1),
c^1(2k) = c^0(2k - 1) + c^1(2k - 1) + c^2(2k - 1),
c^2(2k) = c^2(2k - 1),
c^3(2k) = c^2(2k - 1) + c^3(2k - 1).
For the asymptotic behavior, (c^1(n)+c^2(n))/(c^1(n-1)+c^2(n-1)), (c^0(n)+c^3(n))/(c^0(n-1)+c^3(n-1)) ans a(n)/a(n-1) all tend to 1/2( sqrt(7+ sqrt(5)+ sqrt(38+14 sqrt(5)))) =~ 2.095293985.
Conjectures from Colin Barker, Jan 13 2018: (Start)
G.f.: x*(4 + 3*x - 4*x^2 - x^3) / (1 - x - 3*x^2 + x^3 + x^4).
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4) for n>4.
(End) [Since we have an explicit set of recurrences that produce a(n), it should be straightforward to prove these conjectures. - N. J. A. Sloane, Jan 14 2018]
CROSSREFS
Cf. A131887.
Sequence in context: A116969 A131090 A178615 * A119749 A201498 A145970
KEYWORD
nonn,more
AUTHOR
Shiva Samieinia (shiva(AT)math.su.se), Oct 05 2007, Oct 09 2007
EXTENSIONS
a(11)-a(15) from Neo Scott, Jan 12 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 13 02:27 EDT 2024. Contains 374260 sequences. (Running on oeis4.)