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A061647 Beginning at the well for the topograph of a positive definite quadratic form with values 1, 1, 1 at a superbase (i.e., 1, 1 and 1 are the vonorms of the superbase), these numbers indicate the labels of the edges of the topograph on a path of greatest ascent. 3
1, 3, 9, 23, 61, 159, 417, 1091, 2857, 7479, 19581, 51263, 134209, 351363, 919881, 2408279, 6304957, 16506591, 43214817, 113137859, 296198761, 775458423, 2030176509, 5315071103, 13915036801, 36430039299, 95375081097, 249695203991, 653710530877, 1711436388639 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Form the matrix A=[1,1,1;2,1,0;1,0,0]. a(n) is the sum of the second row elements of A^n. - Paul Barry, Sep 22 2004
REFERENCES
J. H. Conway, The Sensual (Quadratic) Form, MAA.
LINKS
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n > 3, with a(1)=1, a(2)=3, a(3)=9.
a(n) = Fibonacci(n)^2 + Fibonacci(n-1)^2 + 2*Fibonacci(n)*Fibonacci(n+1) + 3*Fibonacci(n-1)*Fibonacci(n), with offset 0. - Paul Barry, Sep 22 2004
a(n) = Fibonacci(n+1)^2 - Fibonacci(n-2)^2 - (-1)^n. - Thomas Baruchel, Jul 29 2005
From J. M. Bergot, Aug 26 2013: (Start)
a(n) = F(n-2)*F(n-1) + F(n-1)*F(n) + F(n)*F(n+1), where F=A000045.
a(n) = (L(2*n+1) + L(2*n-1) + L(2*n-3) - (-1)^n)/5, where L=A000032. [Corrected by Ehren Metcalfe, Mar 26 2016]
Starting at n=2 create Pythagorean triangles by using side x = b^2 - a^2, side y = 2*a*b, and side y = a^2 + b^2. For three successive triangles, let a=F(n) and b=F(n+1), a=F(n+1) and b=F(n+2), and a=F(n+2) and b=F(n+3); then a(n+3)=one-half the sum of the three perimeters.
(End)
G.f.: x*(1+x+x^2) / ( (1+x)*(x^2-3*x+1) ). - R. J. Mathar, Aug 28 2013
a(n) = A001654(n) + A001654(n-1) + A001654(n-2). - R. J. Mathar, Aug 28 2013
a(n) = 4*Fibonacci(n-1)*Fibonacci(n) - (-1)^n. - Bruno Berselli, Oct 30 2015
a(n) = Lucas(2*n-1) - Fibonacci(n-1)*Fibonacci(n). See similar identity for A264080. - Bruno Berselli, Nov 04 2015
a(n) = (1/5)*(4*Lucas(2*n-1) - (-1)^n). - Ehren Metcalfe, Mar 26 2016
a(n) = 2^(-n)*(-(-2)^n - 2*(3-sqrt(5))^n*(1+sqrt(5)) + 2*(-1+sqrt(5))*(3+sqrt(5))^n)/5. - Colin Barker, Sep 30 2016
a(n) = (-1)^(n-1) + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. - Peter Bala, Nov 12 2017
EXAMPLE
a(7) = 417 since a(7) = 2*a(6) + 2*a(5) - a(4) = 2*159 + 2*6 - 23.
MATHEMATICA
LinearRecurrence[{2, 2, -1}, {1, 3, 9}, 40] (* Harvey P. Dale, May 31 2015 *)
PROG
(PARI) x='x+O('x^99); Vec(x*(1+x+x^2)/((1+x)*(x^2-3*x+1))) \\ Altug Alkan, Mar 26 2016
(PARI) a(n) = round(2^(-n)*(-(-2)^n-2*(3-sqrt(5))^n*(1+sqrt(5))+2*(-1+sqrt(5))*(3+sqrt(5))^n)/5) \\ Colin Barker, Sep 30 2016
CROSSREFS
Cf. similar sequences of the type k*F(n)*F(n+1) + (-1)^n listed in A264080.
Sequence in context: A318860 A318818 A026599 * A207008 A077996 A330453
KEYWORD
nonn,easy
AUTHOR
Darrin Frey (freyd(AT)cedarville.edu), Jun 14 2001
STATUS
approved

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)