A215458 a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4) starting 0, 1, 4, 7.

0, 1, 4, 7, 8, 11, 28, 71, 144, 259, 484, 991, 2072, 4187, 8236, 16247, 32544, 65587, 131572, 262543, 523688, 1047179, 2096956, 4196903, 8391600, 16775011, 33546244, 67105087, 134230328, 268455611, 536865868, 1073696471, 2147448384, 4295022739

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277; P1=4, P2=3, Q=2.

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-4).

Formula

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

a(n) = A002248(n)/2.

a(n) = (2^n - (1/2 - (i * sqrt(7))/2)^n - (1/2 + (i * sqrt(7))/2)^n + 1)/2 where i = sqrt(-1). - Paul S. Vanderveen, Jul 08 2017

a(n) = 2^(n-1) - 2^(n/2) * cos(n * arctan(sqrt(7))) + 1/2. - Peter Luschny, Jul 26 2017

Mathematica

CoefficientList[Series[-x (-1 + 2 x^2)/((x - 1) (2*x-1) (2 x^2 - x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)
LinearRecurrence[{4, -7, 8, -4}, {0, 1, 4, 7}, 40] (* Harvey P. Dale, Mar 22 2019 *)

Prog

(Magma) I:=[0, 1, 4, 7]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+8*Self(n-3)-4*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 23 2012
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 8, -7, 4]^n*[0; 1; 4; 7])[1, 1] \\ Charles R Greathouse IV, Jul 07 2017
(Sage)
a = lambda n: (2^n - lucas_number2(n, 1, 2) + 1) // 2
print([a(n) for n in range(34)]) # Peter Luschny, Jul 26 2017

Crossrefs

Sequence in context: A375572 A296029 A000606 * A061932 A270336 A270941

Adjacent sequences: A215455 A215456 A215457 * A215459 A215460 A215461

Keyword

nonn,easy

Author

R. J. Mathar, Aug 11 2012

Status

approved