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%I #5 Oct 05 2015 22:18:22
%S 2,3,4,8,12,16,25,32,36,56,64,96,102,112,128,200,240,256,267,344,356,
%T 384,512,576,598,636,656,936,1024,1040,1048,1128,1344,1368,1480,1952,
%U 2048,2248,2800,2865,2884,2928,3072,3200,3360,3640,4096,4905,5408,5460,5760,6912,6948,7088,7840,8192,8736,9688,10083,11088,11210,11616,13552,14208,15360,16384,16608,18056,18576,20224,21020,22800,24992,31123,31356,32768,33368,33792,34832,35872
%N Union of all unique coefficients of all powers of the g.f. A(x) of this sequence, starting with A(0)=2 and A'(0)=3.
%H Paul D. Hanna, <a href="/A262975/b262975.txt">Table of n, a(n) for n = 0..1024</a>
%e G.f.: A(x) = 2 + 3*x + 4*x^2 + 8*x^3 + 12*x^4 + 16*x^5 + 25*x^6 + 32*x^7 + 36*x^8 + 56*x^9 + 64*x^10 + 96*x^11 + 102*x^12 +...
%e The coefficients in A(x)^n begin:
%e [2, 3, 4, 8, 12, 16, 25, 32, 36, 56, 64, 96, 102, 112, 128, 200, ...];
%e [4, 12, 25, 56, 112, 200, 356, 598, 936, 1480, 2248, 3360, 4905, ...];
%e [8, 36, 102, 267, 636, 1368, 2800, 5460, 10083, 18056, 31356, ...];
%e [16, 96, 344, 1048, 2865, 7088, 16384, 35872, 74584, 148876, ...];
%e [32, 240, 1040, 3640, 11210, 31123, 79940, 193160, 442420, ...];
%e [64, 576, 2928, 11616, 39804, 122148, 344329, 907656, 2260656, ...];
%e [128, 1344, 7840, 34832, 131544, 441532, 1353198, 3858011, ...];
%e [256, 3072, 20224, 99584, 411232, 1497920, 4954608, 15175216, ...];
%e [512, 6912, 50688, 274176, 1229760, 4830624, 17142816, 56099376, ...];
%e [1024, 15360, 124160, 732160, 3546240, 14943872, 56621600, ...]; ...
%e where the sorted union of all unique coefficients forms this sequence.
%e ...
%e The coefficients of A(x)^n are located at the following positions:
%e n=2: [3,5,7,10,14,16,21,25,28,35,38,45,48,53,58,63,69,73,77,85,89,94,102,,...].
%e n=4: [4,9,13,19,26,34,39,50,59,68,75,87,100,112,124,135,149,162,176,189,204,...].
%e n=5: [6,12,20,31,40,54,66,80,95,114,130,152,170,190,209,236,254,275,298,322,...].
%e n=6: [8,17,30,46,61,74,98,120,145,169,196,223,251,285,314,341,377,409,446,476,...].
%e n=7: [11,24,42,62,84,106,138,166,200,238,269,308,349,384,430,470,520,570,625,...].
%e n=8: [18,43,70,104,143,184,234,282,332,385,448,509,576,653,732,808,890,974,...].
%e n=9: [23,52,86,129,177,230,288,345,411,475,554,637,726,815,909,1005,...].
%e n=10: [29,65,108,159,217,280,347,422,495,584,680,777,878,990,...].
%e ...
%e The powers of 2 are located at positions:
%e [1,3,4,6,8,11,15,18,23,29,37,47,56,66,76,91,110,128,150,172,197,224,253,287,321,359,395,437,482,535,591,652,718,783,863,936,1015,...].
%o (PARI) {a(n)=local(A=[2,3],G=A); for(i=1,sqrt(n+1), G=A; for(k=2,4*#binary(n), G=concat(G, Vec(Ser(A)^k)); G=vecsort(G,,8) ); A=Vec(Ser(G) +x*O(x^n)) );A[n]}
%o for(n=1,80,print1(a(n),", "))
%K nonn
%O 0,1
%A _Paul D. Hanna_, Oct 05 2015
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