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A114620 2*A084158 (twice Pell triangles). 1
0, 2, 10, 60, 348, 2030, 11830, 68952, 401880, 2342330, 13652098, 79570260, 463769460, 2703046502, 15754509550, 91824010800, 535189555248, 3119313320690, 18180690368890, 105964828892652, 617608282987020 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Cross-referenced sequences A116484, A001109, A108475, A090390 are also generated by A*B given in the program code.

Related to the reciprocals of the differences between successive convergents of the continued fraction of sqrt(2) (i.e. 1, 2, -10, 60, -348, 2030, -11830, 68952,...). 1/1 + 1/2 - 1/10 + 1/60 - 1/348 + 1/2030 = sqrt(2). 2, 10, 60, ... are products of the denominators of two successive convergents of sqrt(2) (e.g. 11830 = 70*169, cf. A000129 (Pell numbers)). - Gerald McGarvey, Feb 28 2006

a(n) is 1/2 of the even leg(b(n)) of the ordered Pythagorean triple

(x(n),y(n)=x(n)+1,z(n)).b(n)=x(n)+(1-(-1)^n)/2:

x(0)=0,b(0)=0,a(0)=0:x(1)=3,b(1)=4,a(1)=2 . - George F. Johnson, Aug 13 2012

Given a square shape composed of A001110(n+1) number of elements, thinking of it graphically as a sum of layers, each layer having an odd number of elements (all layers together being a sum of consecutive odd numbers), a(n) is the number of last layers that we have to subtract from the square to get a square of squares that is made of [A002965(2*(n+1))]^4 number of elements. - Daniel Poveda Parrilla, Jul 17 2016

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

FORMULA

G.f. 2*x/((x+1)*(x^2-6*x+1))

From George F. Johnson, Aug 13 2012: (Start)

a(n) = ((sqrt(2) + 1)^(2*n+1) - (sqrt(2) - 1)^(2*n+1) - 2*(-1)^n)/8. - corrected by Ilya Gutkovskiy, Jul 18 2016

4*a(n)*(2*a(n)+(-1)^n)+1 = ((A000129(2*n+1))^2 is a perfect square.

n>=0,a(n+1) = 3*a(n) +(-1)^n +sqrt(4*a(n)*(2*a(n)+(-1)^n)+1).

n>0 ,a(n-1) = 3*a(n) +(-1)^n -sqrt(4*a(n)*(2*a(n)+(-1)^n)+1).

a(n+1) = 6*a(n) -a(n-1) +2*(-1)^n : a(n+1) = 5*a(n)+5*a(n-1)-a(n-2).

n>0,a(n+1)*a(n-1)=a(n)*(a(n)+2*(-1)^n):a(n)=(A046729(n))/2

(End)

a(n) = A000129(n)*A000129(n+1). - Philippe Deléham, Apr 10 2013

a(n) = A002965(2*(n+1))*[A002965(2*(n+1)+1) - A002965(2*(n+1))]. - Daniel Poveda Parrilla, Jul 17 2016

MATHEMATICA

Table[Fibonacci[n, 2] Fibonacci[n + 1, 2], {n, 0, 20}] (* or *)

LinearRecurrence[{5, 5, -1}, {0, 2, 10}, 21] (* or *)

CoefficientList[Series[2 x/((x + 1) (x^2 - 6 x + 1)), {x, 0, 20}], x] (* Michael De Vlieger, Jul 17 2016 *)

PROG

Floretion Algebra Multiplication Program, FAMP Code: 1jesleftseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'

CROSSREFS

Cf. A116484, A001109, A108475, A090390.

Cf. A000129.

Sequence in context: A303361 A026161 A025188 * A173613 A004981 A214764

Adjacent sequences:  A114617 A114618 A114619 * A114621 A114622 A114623

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Feb 17 2006

STATUS

approved

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Last modified October 30 05:34 EDT 2020. Contains 338077 sequences. (Running on oeis4.)