A083353 - OEIS

%I #15 Nov 19 2017 14:20:41

%S 1,9,36,81,144,225,324,576,1089,1296,1764,2916,4356,6084,7569,9801,

%T 16641,20736,25281,32400,39204,53361,69696,76176,90000,110889,149769,

%U 176400,184041,207936,281961,338724,363609,349281,455625,540225,665856,772641,746496,893025,1258884,1375929,1565001,1587600,1782225,2259009,2576025,2722500,2985984,3186225,3837681,4284900,4928400,5143824,5475600,6320196,7290000,8311689,8520561,8803089

%N Coefficients of power series A(x) consist entirely of squares, where A(x) = A083352(x)^2 + A083352(x) - 1.

%C After the first term, all terms seem to be a multiples of 9.

%H Paul D. Hanna, <a href="/A083353/b083353.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + 9*x + 36*x^2 + 81*x^3 + 144*x^4 + 225*x^5 + 324*x^6 + 576*x^7 + 1089*x^8 + 1296*x^9 + 1764*x^10 + 2916*x^11 + 4356*x^12 + 6084*x^13 + 7569*x^14 + 9801*x^15 + 16641*x^16 + 20736*x^17 + 25281*x^18 + 32400*x^19 + 39204*x^20 + 53361*x^21 + 69696*x^22 + 76176*x^23 + 90000*x^24 + 110889*x^25 + 149769*x^26 + 176400*x^27 + 184041*x^28 + 207936*x^29 + 281961*x^30 + ...

%e where A(x) = B(x)^2 + B(x) - 1 with B(x) the g.f. of A083352, which begins:

%e B(x) = 1 + 3*x + 9*x^2 + 9*x^3 + 3*x^4 + 15*x^5 + 33*x^6 + 18*x^7 + 36*x^8 + 24*x^9 + 75*x^10 + 96*x^11 + 51*x^12 + 96*x^13 + 159*x^14 + 165*x^15 + 255*x^16 + 168*x^17 + 27*x^18 + 60*x^19 + 333*x^20 + 255*x^21 + 66*x^22 + 18*x^23 + 441*x^24 + 291*x^25 + 735*x^26 + 258*x^27 + 390*x^28 + 696*x^29 + 480*x^30 + ...

%o (PARI) {for(i=1, 30, A=[1]; print1(1, ", "); for(i=1, 200, A0=concat(A, 0); for(n=1, 100*A[#A], A0[#A0]=n; B=Vec(Ser(A0)^2 + Ser(A0) - 1); if(issquare(B[#B]), print1(B[#B], ", "); A0[#A0]=n; A=A0; break)))); B}

%Y Cf. A083352, A083354 (square root of terms).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 26 2003

%E Extended by _Paul D. Hanna_, Nov 19 2017