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A061646 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) with a(-1) = 1, a(0) = 1, a(1) = 1. 14
1, 1, 1, 3, 7, 19, 49, 129, 337, 883, 2311, 6051, 15841, 41473, 108577, 284259, 744199, 1948339, 5100817, 13354113, 34961521, 91530451, 239629831, 627359043, 1642447297, 4299982849, 11257501249, 29472520899, 77160061447, 202007663443, 528862928881 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,4

COMMENTS

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 values of the quadratic form at vectors adjacent to the path in the topograph of greatest rate of ascent of labels of the edges of the topograph.

For n>1, a_n is the number of domino tilings of the L-grid obtained by removing the upper-right (n-1)x(n-2) rectangle from a (n+1)xn rectangle; also, for n>1, (2*a_n)^2 is the number of domino tilings of the holey square obtained by removing the centered (n-2)x(n-2) square from a (n+2)x(n+2) square. - Roberto Tauraso, Jun 05 2004

Let P denote the 3 X 3 Fibonacci matrix [ 0 0 1 / 0 1 2 / 1 1 1 ]. Then a(n) is the central term of P^n. - Gary W. Adamson, May 13 2003

Coefficient of 1 when looking for the simplest linear dependence between (phi^(n-1)+fibonacci(n-1)) / (phi^n + fibonacci(n)) - 1/phi, 1 and phi. Thus a(n) is given by lindep([phi^(n-1)+fibonacci(n-1))/(phi^n+fibonacci(n))-1/phi,1,phi],80)[2] when using Pari-GP with enough digits of precision. - Thomas Baruchel, Nov 19 2004

a(n), n>=2, is twice the area of the plane triangle in three dimensional space with vertices (F(n-1),0,0), (0,F(n),0) and (0,0,F(n+1)). See the Atanassov et al. reference p. 88 (misprint in eq. (1.3): it should read F_{2n-1} not (F_{2n-1})^2). - Wolfdieter Lang, Jul 22 2005.

From L. Edson Jeffery, Apr 20 2011: (Start):

Let U be the unit-primitive matrix (see [Jeffery])

U=U_(10,2)=

(0 0 1 0 0)

(0 1 0 1 0)

(1 0 1 0 1)

(0 1 0 2 0)

(0 0 2 0 1).

Then a(n) = (Trace(U^n))/5 = (U^n)_(5,5) = ((U^n)_(1,1)+(U^n)_(4,4))/2 = ((U^n)_(2,2)+(U^n)_(3,3))/2, n>=0, recalling that the offset for A061646 is -1. (See also A189316.) (End)

REFERENCES

J. H. Conway, The Sensual (Quadratic) Form, MAA.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = -1..1000

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017.

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.7.

K. T. Atanassov, V. Atanassova, A. G. Shannon and J. C. Turner, New visual perspectives on Fibonacci numbers, World Scientific, 2002.

Jonny Griffiths and Martin Griffiths, Fibonacci-related sequences via iterated QRT maps, Fib. Q., 51 (2013), 218-227.

Theodore Hwa, Posting to sci.math.

L. E. Jeffery, Unit-primitive matrices.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001; pages 383-384.

Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.

Roberto Tauraso, A New Domino Tiling Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.3.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

FORMULA

a(n) = Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2 = A007598(n+1) - A001654(n-1). Area of triangle with sides sqrt(a(n)), sqrt(a(n-1)) and sqrt(a(n-2)) is sqrt(3)/4, i.e. 2*(a(n)*a(n-1) + a(n)*a(n-2)  + a(n-1)*a(n-2)) - (a(n)^2 + a(n-1)^2 + a(n-2)^2) = 3. - Henry Bottomley, Jan 09 2003

a(n) = (2*fibonacci(n)*fibonacci(n+1)*(fibonacci(n+2) + phi*fibonacci(n+1)) + (1/phi)^n)/(fibonacci(n)*phi+fibonacci(n+1)). - Thomas Baruchel, Nov 19 2004

a(n) = F(2*n-1) + F(n-1)*F(n), with F(-3):=2, F(-2):=-1 and F(-1):=1 (corrected eq. (1.3) of the Atanassov et al. reference) with F(n):=A000045 (Fibonacci). - Wolfdieter Lang, Jul 22 2005

G.f.: 1/x + (1-x-x^2)/((1+x)*(1-3x+x^2)). - Philippe Deléham, Dec 16 2008

a(n) = F(n)*F(n+1) + F(n-1)^2 = A001654(n) + A007598(n-1), n>=-1.

a(n) = 2*F(n)^2 + (-1)^n = A175395(n) + (-1)^n, n>-1.

a(n) = 1/5*Sum_{k=1..5) ((w_k)^2-1)^n, w_k=2*cos((2*k-1)*Pi/10), n>=0, recalling that the offset for A061646 is -1. - L. Edson Jeffery, Apr 20 2011.

a(n) = ((-2)^n+2*(3-Sqrt(5))^n + 2*(3+Sqrt(5))^n)/(5*2^n). - L. Edson Jeffery, Apr 21 2011

a(n+1)-a(n) = 2*A001654(n+2). - J. M. Bergot, Jun 12 2013

a(n) = L(2*n-2) + F(n-3)*F(n-2) for n>=3, where L=A000032 are Lucas numbers. - J. M. Bergot, Aug 08 2012

a(n-1)+a(n) = A052995(n) for n>0. - R. J. Mathar, Aug 15 2013

For n>=1, a(n) = (F(n+2)^3 + F(n-2)^3)/(9*F(n)).  - Richard R. Forberg, Nov 17 2014

EXAMPLE

a(7)=337 since 2*a(6)+2*a(5)-a(4) = 2*129+2*49-19 = 337.

MAPLE

with (combinat):

F:= n-> fibonacci(n):

seq (F(n)*F(n+1) +F(n-1)^2, n=-1..27);

seq(2*F(n)^2+(-1)^n, n=0..27);

MATHEMATICA

LinearRecurrence[{2, 2, -1}, {1, 1, 1}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

PROG

(Haskell)

a061646 n = a061646_list !! (n + 1)

a061646_list = 1 : 1 : 1 : zipWith (-) (map (* 2)

   (zipWith (+) (drop 2 a061646_list) (tail a061646_list))) a061646_list

-- Reinhard Zumkeller, Dec 02 2011

(PARI) a(n)=fibonacci(n)^2+fibonacci(n)*fibonacci(n-1)+fibonacci(n-1)^2 \\ Charles R Greathouse IV, Jun 13 2013

(MAGMA) I:=[1, 1, 1]; [n le 3 select I[n] else 2*Self(n-1) +2*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 07 2019

(Sage) [2*fibonacci(n)^2 + (-1)^n for n in (-1..30)] # G. C. Greubel, Jan 07 2019

(GAP) a:=[1, 1, 1];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 07 2019

CROSSREFS

Sequence in context: A007288 A191824 A191757 * A017926 A017927 A116903

Adjacent sequences:  A061643 A061644 A061645 * A061647 A061648 A061649

KEYWORD

nonn,easy

AUTHOR

Darrin Frey (freyd(AT)cedarville.edu), Jun 14 2001

STATUS

approved

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)