A182193 Sequence of row differences related to table A182355.

-1, 1, 19, 125, 743, 4345, 25339, 147701, 860879, 5017585, 29244643, 170450285, 993457079, 5790292201, 33748296139, 196699484645, 1146448611743, 6681992185825, 38945504503219, 226991034833501, 1323000704497799, 7711013192153305, 44943078448422043

Offset

0,3

Comments

Sequence of row differences in table A182355. If A182355(k + 1, 0) - A182355(k, 0) = -1, a(n) = A182355(k + 1, n) - A182355(k, n).

If p is a prime of the form 8r = +/- 3, a(p) = 5 mod p; if p is a prime of the form 8r = +/- 1, a(p) = 1 mod p.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Formula

a(n) = 6*a(n-1) - a(n-2) + 12.

a(0)=-1, a(1)=1, a(2)=19, a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - Harvey P. Dale, Feb 09 2014

From Colin Barker, Mar 05 2016: (Start)

a(n) = -3 + (1/4)*( (4-sqrt(2))*(3+2*sqrt(2))^n + (4+sqrt(2))*(3-2*sqrt(2))^n ).

G.f.: -(1-8*x-5*x^2) / ((1-x)*(1-6*x+x^2)).

(End)

a(n) = A002203(2*n) - A000129(2*n) - 3. - G. C. Greubel, May 24 2021

Maple

Pell:= proc(n) option remember;
    if n<2 then n
  else 2*Pell(n-1) + Pell(n-2)
    fi; end:
seq(Pell(2*n) + 2*Pell(2*n-1) - 3, n=0..40); # G. C. Greubel, May 24 2021

Mathematica

LinearRecurrence[{7, -7, 1}, {-1, 1, 19}, 30] (* Harvey P. Dale, Feb 09 2014 *)

Prog

(Magma) I:=[-1, 1]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..30]]; // Vincenzo Librandi, Feb 10 2014
(PARI) Vec(-(1-8*x-5*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 05 2016
(Sage) [lucas_number2(2*n, 2, -1) - lucas_number1(2*n, 2, -1) - 3 for n in (0..40)] # G. C. Greubel, May 24 2021

Crossrefs

Cf. A000129, A002203, A182188, A182189, A182191, A182355.

Sequence in context: A125329 A126487 A241965 * A109669 A164905 A142106

Adjacent sequences: A182190 A182191 A182192 * A182194 A182195 A182196

Keyword

sign,easy

Author

Kenneth J Ramsey, Apr 17 2012

Extensions

More terms from Harvey P. Dale, Feb 09 2014

Status

approved