A131515 - OEIS

%I #19 Aug 05 2016 10:39:45

%S 1,16,545,18516,629001,21367520,725866681,24658099636,837649520945,

%T 28455425612496,966646821303921,32837536498720820,1115509594135203961,

%U 37894488664098213856,1287297104985204067145,43730207080832840069076,1485539743643331358281441

%N a(n+2) = 34*a(n+1) - a(n) + 2; a(1) = 1 and a(2) = 16.

%H Colin Barker, <a href="/A131515/b131515.txt">Table of n, a(n) for n = 1..650</a>

%H Chip Curtis, <a href="http://www.jstor.org/stable/27646512">Generating Heronian Triangles</a>, The College Mathematics Journal, vol. 38 (2007), pp. 315-316. See page 316.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).

%F G.f.: -x*(1-19*x+20*x^2)/(-1+x)/(1-34*x+x^2). - _R. J. Mathar_, Nov 14 2007

%F a(1)=1, a(2)=16, a(3)=545, a(n) = 35*a(n-1)-35*a(n-2)+a(n-3). - _Harvey P. Dale_, Feb 10 2015

%F a(n) = (-6 + (963-680*sqrt(2))*(17+12*sqrt(2))^n + (17+12*sqrt(2))^(-n)*(963+680*sqrt(2)))/96. - _Colin Barker_, Mar 02 2016

%e If n=3 then a(3) = 34*a(2) - a(1) + 2 = 545 which is the third term in the sequence.

%t RecurrenceTable[{a[1]==1,a[2]==16,a[n]==34a[n-1]-a[n-2]+2},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{1,16,545},20] (* _Harvey P. Dale_, Feb 10 2015 *)

%o (PARI) Vec(x*(1-19*x+20*x^2)/((1-x)*(1-34*x+x^2)) + O(x^20)) \\ _Colin Barker_, Mar 02 2016

%o (PARI) a(n)=([0,1,0;0,0,1;1,-35,35]^n*[20;1;16])[1,1] \\ _Charles R Greathouse IV_, Aug 05 2016

%Y Cf. A016064, A001110.

%K nonn,easy

%O 1,2

%A _Parthasarathy Nambi_, Aug 14 2007

%E Definition corrected by _Rick L. Shepherd_, Aug 17 2007

%E More terms from _Harvey P. Dale_, Feb 10 2015