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A080239
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Diagonal sums of triangle A035317.
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8
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1, 1, 2, 3, 6, 9, 15, 24, 40, 64, 104, 168, 273, 441, 714, 1155, 1870, 3025, 4895, 7920, 12816, 20736, 33552, 54288, 87841, 142129, 229970, 372099, 602070, 974169, 1576239, 2550408, 4126648
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Convolution of Fibonacci sequence with sequence (1,0,0,0,1,0,0,0,1, ...)
There is an interesting relation between a(n) and the Fibonacci sequence f(n). Sqrt(a(4n-2)) = f(2n), where we denote by Sqrt() the square root. By using this fact we can calculate the value of a(n) by the following (1),(2),(3),(4) and (5). (1) a(1) = 1. (2) If n = 2 (mod 4), then a(n) = f((n+2)/2)^2. (3) If n = 3 (mod 4), then a(n) = (f((n+5)/2)^2-2f((n+1)/2)^2-1)/3. (4) If n = 0 (mod 4), then a(n) = (f((n+4)/2)^2+f(n/2)^2-1)/3. (5) If n = 1 (mod 4), then a(n) = (2f((n+3)/2)^2-f((n-1)/2)^2+1)/3. - Hiroshi Matsui and Ryohei Miyadera (miyadera1272000(AT)yahoo.co.jp), Aug 08 2006
Sequences of the form s(0)=a,s(1)= b, s(n) =s(n-1)+s(n-2)+k if n mod m = p,else s(n)=s(n-1)+s(n-2) will have a form fib(n-1)*a+fib(n)*b+P(x)*k. a(n) is the P(x) sequence for m= 4...s(n)=fib(n)*a + fib(n-1)*b +a(n-4-p)*k [From Gary Detlefs (gdetlefs(At)aol.com) Dec 05 2010]
A different formula for a(n) as a function of the Fibonacci numbers f(n) may be conjectured. The pattern is of the form a(n)=f(p)*f(p-q) -1 if n mod 4 = 3 , else f(p)*f(p-q) where p = 2,2,4,4,4,4,6,6,6,6,8,8,8,8... and q = 0,1,3,2,0,1,3,2,0,1,3,2... p(n) = 2* A002265(n+4) = 2*(floor((n+3)/2)-floor((n+3)/4)) (see Von Post Comment). A general formula for sequences having period 4 with terms a,b,c,d is given in A121262 (the discrete Fourier transform, as for all periodic sequences) and is a function of t(n)= 1/4*(2*cos(n*Pi/2)+1+(-1)^n). r4(a,b,c,d,n) = a*t(n+3)+b*t(n+2)+c*t(n+1)+d*t(n). This same formula may be used to subtract the 1 at n mod 4 = 3.
a(n)= f(p(n))*f(p(n)-r4(1,0,3,2,n))- r4(0,0,1,0,n). [Gary Detlefs, Dec 09 2010]
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REFERENCES
| H. Matsui et al., Problem B-1019, Fibonacci Quarterly, Vol. 45, Number 2; 2007; p. 182.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index to sequences with linear recurrences with constant coefficients, signature (1,1,0,1,-1,-1).
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FORMULA
| G.f.: x/((1-x^4)(1-x-x^2)) = x/(1-x-x^2-x^4+x^5+x^6); a(n)=a(n-1)+a(n-2)+a(n-4)-a(n-5)-a(n-6).
a(n)=Sum{j=0..floor(n/2) Sum{k=0..floor((n-j)/2) binomial(n-j-2k, j-2k}}.
Another recurrence is given in the Maple code.
If n mod 4 = 1 then a(n)=a(n-1)+a(n-2)+1, else a(n)= a(n-1)+a(n-2). [From Gary Detlefs (gdetlefs(at)aol.com) Dec 05 2010]
a(4n)= A058038(n) = Fibonacci(2n+2)*Fibonacci(2n).
a(4n+1) = A081016(n) = Fibonacci(2n+2)*Fibonacci(2n+1).
a(4n+2) = A049682(n+1) = Fibonacci(2n+2)^2.
a(4n+3) = A081018(n+1) = Fibonacci(2n+2)*Fibonacci(2n+3).
a(n)=8*a(n-4)-8*a(n-8)+a(n-12), n>12 [From Gary Detlefs (gdetlefs(at)aol.com) Dec 10 2010]
a(n+1) = a(n) + a(n-1) + A011765(n+1). [Reinhard Zumkeller, Jan 06 2012]
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MAPLE
| f:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 4 = 1 then t1:=1 else t1:=0; fi: f(n-1)+f(n-2)+t1; end; [seq(f(n), n=1..100)]; - N. J. A. Sloane (njas(AT)research.att.com), May 25 2008
with(combinat): f:=n-> fibonacci(n): p:=n-> 2*(floor((n+3)/2)-floor((n+3)/4)): t:=n-> 1/4*(2*cos(n*Pi/2)+1+(-1)^n): r4:=(a, b, c, d, n)-> a*t(n+3)+b*t(n+2)+c*t(n+1)+d*t(n): seq(f(p(n)*f(p(n)-r4(1, 0, 3, 2, n))-r4(0, 0, 1, 0, n), n= 1..33); # Gary Detlefs, Dec 09 2010
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MATHEMATICA
| (*f[n] is the Fibonacci sequence and a[n] is the sequence of A080239*) f[n_] := f[n] = f[n - 1] + f[n - 2]; f[1] = 1; f[2] = 1; a[n_] := Which[n == 1, 1, Mod[n, 4] == 2, f[(n + 2)/2]^2, Mod[n, 4] == 3, (f[(n + 5)/2]^2 - 2f[(n + 1)/2]^2 - 1)/3, Mod[n, 4] == 0, (f[(n + 4)/2]^2 + f[n/2]^2 - 1)/3, Mod[n, 4] == 1, (2f[(n + 3)/2]^2 - f[(n - 1)/2]^2 + 1)/3] - Hiroshi Matsui and Ryohei Miyadera (miyadera1272000(AT)yahoo.co.jp), Aug 08 2006
a=0; b=0; lst={a, b}; Do[z=a+b+1; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z, {n, 4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 16 2010]
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PROG
| (Haskell)
a080239 n = a080239_list !! (n-1)
a080239_list = 1 : 1 : zipWith (+)
(tail a011765_list) (zipWith (+) a080239_list $ tail a080239_list)
-- Reinhard Zumkeller, Jan 06 2012
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CROSSREFS
| Cf. A035317, A000045, A026636, A026647, A004695.
Sequence in context: A014214 A094993 A192671 * A114323 A018158 A057928
Adjacent sequences: A080236 A080237 A080238 * A080240 A080241 A080242
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 11 2003
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