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A221212 Rounded area of distinct right triangles appearing in the unit golden spiral. 1
1, 3, 4, 7, 9, 18, 25, 47, 65, 123, 170, 322, 445, 843, 1165, 2207, 3050, 5778, 7985, 15127, 20905, 39603, 54730, 103682, 143285, 271443, 375125, 710647, 982090, 1860498, 2571145, 4870847, 6731345, 12752043, 17622890, 33385282, 46137325, 87403803, 120789085 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = round(x(n)), where x(1) = 1, x(3) = phi + 1 and phi is the golden ratio. For n >= 1, x(2n) = F(2n)*phi + F(2n-1) and for n >= 2, x(2n+1) = x(2n) + x(2n-2).
LINKS
FORMULA
Conjecture: a(n) = 3*a(n-2)-a(n-4) for n>5. G.f.: -x*(x+1)*(x^7-x^6-x^5+x^4+x^3+x^2-2*x-1) / ((x^2-x-1)*(x^2+x-1)). - Colin Barker, Mar 12 2013
MATHEMATICA
m=100; x[1]=1; x[2]=GoldenRatio+1; x[3]=GoldenRatio+2; Do[x[n]=If[EvenQ[n], Fibonacci[n]*GoldenRatio+Fibonacci[n-1], x[n-1]+x[n-3]], {n, 4, m}]; Table[Round[x[n]], {n, m}] (* Zak Seidov, Feb 23 2013 *)
PROG
(Small Basic)
x[1]=1
f[1]=1
f[2]=1
For n = 3 To 50
f[n]=f[n-1]+f[n-2] 'Fibonacci
EndFor
For nn = 2 To 50
If Math.Remainder(nn, 2)=0 Then 'Even terms
x[nn]=f[nn]*((1+Math.SquareRoot(5))/2)+f[nn-1]
Else
If nn = 3 then
x[nn]=x[nn-1]+x[nn-2] 'Odd terms
EndIf
If nn >= 5 Then
x[nn]=x[nn-1]+x[nn-3]
EndIf
EndIf
EndFor
For j = 1 to 50
TextWindow.Write(j+" ")
TextWindow.Write(math.Round(x[j]))
TextWindow.WriteLine(" ")
EndFor
CROSSREFS
Even terms of a(n) are A005248.
Sequence in context: A048849 A076211 A167186 * A218278 A101062 A359394
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Feb 21 2013
STATUS
approved

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Last modified March 29 22:15 EDT 2024. Contains 371282 sequences. (Running on oeis4.)