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A182193
Sequence of row differences related to table A182355.
2
-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.
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
KEYWORD
sign,easy
AUTHOR
Kenneth J Ramsey, Apr 17 2012
EXTENSIONS
More terms from Harvey P. Dale, Feb 09 2014
STATUS
approved