Challenge Secuencia Recaman

In mathematics and computer science, the Recamán’s sequence (or Recaman’s sequence) is a well known sequence defined by a recurrence relation, because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.

It takes its name after its inventor Bernardo Recamán Santos (Bogotá, August 5, 1954), a Colombian mathematician.

(taken from Wikipedia: https://en.wikipedia.org/wiki/Recam%C3%A1n%27s_sequence)

Recaman Challenge

Numberphile (youtube channel) challenged viewers to replicate Recaman Sequence graph, and have the output be:

Video can be found : https://www.youtube.com/watch?v=FGC5TdIiT9U Challenge is located in min 4 sec 30 until min 4 sec 41

Creating Recaman Sequence

funcionSecuenciaRecaman1<-function(numero) { 
  x<- vector()
  y<- vector()
  z<-0
  i <- 1
  
  while(i <= numero)
  {
    x[i] <- z
    
    if(i == 1){
      
      y[i] <- z
      
    } else {
      
      if(y[i-1] - x[i] > 0 & is.na(match(y[i-1] - x[i],y))) {
        y[i] <- y[i-1] - x[i]
      }  else {
        y[i] <- y[i-1] + x[i]
      }
      
    }
    
    z<- z+1
    i = i+1
  }
  
  return(cbind(x,y))  
}

Let’s test it out:

funcionSecuenciaRecaman1(25)

##        x  y
##  [1,]  0  0
##  [2,]  1  1
##  [3,]  2  3
##  [4,]  3  6
##  [5,]  4  2
##  [6,]  5  7
##  [7,]  6 13
##  [8,]  7 20
##  [9,]  8 12
## [10,]  9 21
## [11,] 10 11
## [12,] 11 22
## [13,] 12 10
## [14,] 13 23
## [15,] 14  9
## [16,] 15 24
## [17,] 16  8
## [18,] 17 25
## [19,] 18 43
## [20,] 19 62
## [21,] 20 42
## [22,] 21 63
## [23,] 22 41
## [24,] 23 18
## [25,] 24 42

Creating Challenge Plot

plottingChallenge<-function(numeral){

#Recaman Sequence: (only retrieving values)
laSecuencia <-funcionSecuenciaRecaman1(numeral)[,2]


#declarandoVectoresNecesarios
puntosEnx <- vector()
puntosEny <- vector()
ciclo <- vector()
degrees <- c(1:179) #to simulate semi-circle we will need degrees from 1 : 179
numeral2 <- numeral - 1
#Comienza el Ciclo

for(i in 1:numeral2)
{
  ##These are the edges of the semi-circle (diameter of circle) 
    x1<- laSecuencia[i] 
    x2<- laSecuencia[i+1]

  ##Evaluating whether semi-circle will curve upwards or downwards  
    if(i%%2 == 0){curvaturaHaciaArriba <- 1} else {curvaturaHaciaArriba <- -1}
    
    radio <- abs(x2 - x1)/2 #radius of circle
    if(x2>x1){puntoMedio <- x1+radio}else{puntoMedio <- x2+radio }
    
    alturas<- sin(degrees*pi/180)*radio*curvaturaHaciaArriba #all heights
    distancias <-  cos(degrees*pi/180)*radio #all lengths
    
    if(x2>x1){distancias1 <- sort(distancias,decreasing = F)
    } else {distancias1 <- distancias}
    
    puntosEnx <- c(puntosEnx,x1,distancias1 + puntoMedio,x2)
    puntosEny <- c(puntosEny,0,alturas,0)
    
}

matriz<-cbind(puntosEnx,puntosEny)
matrizUnica<-unique(matriz)


par(bg="black")
plot(matrizUnica[,1],matrizUnica[,2], type = "l", 
     main = "Secuencia de Recaman", 
           xlab = "Seq. Recaman", 
           ylab = "",
     col="white", #color del grafico
     #col.axis = "white", #estos son los numeritos,
     col.axis = "yellow", #titulos de los axis
     cex.lab = 1
           
     )
points( tail(matrizUnica[,1],1), tail(matrizUnica[,2],1), pch = 11, col = "yellow"  )
title("Recaman Sequence", col.main = "white")
}

Testing it out:

plottingChallenge(62)