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A214525 a(n) = 7*a(n-1) - 23*a(n-2) + 49*a(n-3) - 49*a(n-4) with a(0)=0, a(1)=1, a(2)=7, a(3)=19.

%I A214525 #44 Dec 23 2024 14:53:43
%S A214525 0,1,7,19,21,4,133,937,2667,3439,2128,20569,132867,392743,596869,
%T A214525 647596,3539109,19881229,60254719,106198903,158297664,643809889,
%U A214525 3117087967,9564827611,19050869061,34555674196,119658973525,507648339217,1561117435059,3421971910543
%N A214525 a(n) = 7*a(n-1) - 23*a(n-2) + 49*a(n-3) - 49*a(n-4) with a(0)=0, a(1)=1, a(2)=7, a(3)=19.
%C A214525 This is a divisibility sequence.
%C A214525 It factors over the Eisenstein-Jacobi integers into two 2nd order sequences (with w^3 = 1): 0, 1, w+3, 3w+5, 4w+5, 2, -12w-1, -29w+3, ...  and its conjugate (replace  w  by  w^2).  The relation for this is a(n) = (w+3)a(n-1) - (2w+3)a(n-2).
%H A214525 Bruno Berselli, <a href="/A214525/b214525.txt">Table of n, a(n) for n = 0..1000</a>
%H A214525 R. K. Guy, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-July/001955.html">New sequence?</a>, SeqFan, 2009
%H A214525 Hugh Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory, 7(5) (2011), 1255-1277.
%H A214525 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-23,49,-49).
%F A214525 G.f.: x*(1-7*x^2) /(1-7*x+23*x^2-49*x^3+49*x^4). - _Bruno Berselli_, Aug 08 2012
%t A214525 RecurrenceTable[{a[0] == 0, a[1] == 1, a[2] == 7, a[3] == 19, a[n] == 7 a[n - 1] - 23 a[n - 2] + 49 a[n - 3] - 49 a[n - 4]}, a[n], {n, 0, 29}] (* _Bruno Berselli_, Aug 08 2012 *)
%t A214525 LinearRecurrence[{7,-23,49,-49},{0,1,7,19},30] (* _Harvey P. Dale_, Jan 02 2023 *)
%K A214525 nonn,easy
%O A214525 0,3
%A A214525 _N. J. A. Sloane_, Aug 07 2012, based on a posting to the Sequence Fans Mailing List by _R. K. Guy_, Jul 29 2009
%E A214525 a(9) corrected by _Bruno Berselli_, Aug 08 2012