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%I #22 Mar 24 2024 04:24:42
%S 1,16,271,4591,77776,1317601,22321441,378146896,6406175791,
%T 108526841551,1838550130576,31146825378241,527657481299521,
%U 8939030356713616,151435858582831951,2565470565551429551,43461563755791470416,736281113282903567521,12473317362053569177441
%N The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.
%C The 2 equations are equivalent to the Pell equation x^2- 285*y^2=1,
%C with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
%C Also: the first differences of A078366.
%C Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - _Colin Barker_, Feb 14 2014
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,-1).
%F a(t+2) = 17*a(t+1)-a(t).
%F a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).
%F a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.
%F G.f.: x*(1-x)/(1-17*x+x^2).
%F a(n) = 17*a(n-1)-a(n-2). - _Colin Barker_, Feb 14 2014
%p t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
%p if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t,a,b,n): end if: end do:
%t Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2),{x,0,40}],x]] (* or *) LinearRecurrence[{17,-1},{1,16},20] (* _Harvey P. Dale_, Oct 12 2012 *)
%o (PARI) Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 14 2014
%Y Cf. A078366, A160682, A161599 (sequence of B), A161583 (sequence of k).
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 1,2
%A _Paul Weisenhorn_, Jun 14 2009
%E Edited, extended by _R. J. Mathar_, Sep 02 2009
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