A271972 Expansion of (1 + 3*x)/(1 - 4*x + 7*x^2).

1, 7, 21, 35, -7, -273, -1043, -2261, -1743, 8855, 47621, 128499, 180649, -176897, -1972131, -6650245, -12796063, -4632537, 71042293, 316596931, 769091673, 860188175, -1942889011, -13792873269, -41571269999, -69734967113, 12059021541, 536380855955, 2061110273033, 4489775100447

Offset: 0

Author

Ilya Gutkovskiy, Jul 13 2016

Keywords

• sign
• easy

Comments

Satisfies of recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=1, b(0)=2.

More generally, for the recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=k, b(0)=m solution is a(n) = ((2 + i*sqrt(3))^n*((sqrt(3) - i)*k - 2*i*m) + (2 - i*sqrt(3))^n*((sqrt(3) + i)*k + 2*i*m))/(2*sqrt(3)), b(n) = ((2 - i*sqrt(3))^n*((sqrt(3) - i)*m - 2*i*k) + (2 + i*sqrt(3))^n*(2*i*k + (sqrt(3) + i)*m))/(2*sqrt(3)), where i is the imaginary unit.

Convolution of A169585 and A168175.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-7)

Crossrefs

Cf. A168175, A169585.

Programs

Maple

a:=series((1+3*x)/(1-4*x+7*x^2),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Mar 27 2019

Mathematica

LinearRecurrence[{4, -7}, {1, 7}, 30]

PARI

Vec((1+3*x)/(1-4*x+7*x^2) + O(x^99)) \\ Altug Alkan, Jul 13 2016

Formula

O.g.f.: (1 + 3*x)/(1 - 4*x + 7*x^2).

E.g.f.: (5*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3.

a(n) = 4*a(n-1) - 7*a(n-2).

a(n) = ((2 + i*sqrt(3))^n*(-5*i + sqrt(3)) + (2 - i*sqrt(3))^n*(5*i + sqrt(3)))/(2*sqrt(3)), where i is the imaginary unit.

a(n) = 7^(n/2)*((5/sqrt(3))*sin(c)+cos(c)) with c = n*arctan(sqrt(3)/2). - Peter Luschny, Jul 21 2016