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A083352
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Least positive integer coefficients of power series A(x) such that the coefficients of A(x)^2 + A(x) - 1 consist entirely of squares.
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3
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1, 3, 9, 9, 3, 15, 33, 18, 36, 24, 75, 96, 51, 96, 159, 165, 255, 168, 27, 60, 333, 255, 66, 18, 441, 291, 735, 258, 390, 696, 480, 696, 1062, 423, 927, 681, 867, 1731, 300, 1131, 969, 1959, 222, 1812, 1047, 2487, 3027, 1020, 537, 2859, 1788, 1698, 63, 2712, 2250, 2934, 282, 2409, 1533, 3243, 1188, 7002, 1779, 1038, 207, 4110, 6156, 7068, 7470, 825, 3501, 3612, 6984, 9945, 2733, 7383, 1581, 2730, 2616, 7476, 13959, 13224, 15078, 12837, 2979, 2805, 2943, 15426, 8871, 4536
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OFFSET
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0,2
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COMMENTS
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After the first term, all terms seem to be multiples of 3.
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LINKS
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EXAMPLE
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G.f.: A(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 + ...
Let the B(x) denote the g.f. of A083353, then
B(x) = A(x)^2 + A(x) - 1 = 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 + ...
the coefficients of which are squares (see A083354).
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PROG
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(PARI) {for(i=1, 30, A=[1]; print1(A[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(n, ", "); A0[#A0]=n; A=A0; break)))); A}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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