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A289980 Oblong (or pronic) numbers of the form x_n.y_n where z_n indicates the digit z repeated n times.

%I A289980 #37 Dec 02 2018 11:10:30
%S A289980 12,20,30,42,56,72,90,1122,4422,9900,111222,444222,999000,11112222,
%T A289980 44442222,99990000,1111122222,4444422222,9999900000,111111222222,
%U A289980 444444222222,999999000000,11111112222222,44444442222222,99999990000000,1111111122222222
%N A289980 Oblong (or pronic) numbers of the form x_n.y_n where z_n indicates the digit z repeated n times.
%C A289980 There are only three infinite subsequences, that is, x_n.y_n is pronic for all n, and they occur with (x,y) = (1,2), (4,2), or (9,0). (See Crux Mathematicorum link.)
%H A289980 Vincenzo Librandi, <a href="/A289980/b289980.txt">Table of n, a(n) for n = 1..300</a>
%H A289980 Charles W. Trigg, <a href="https://cms.math.ca/crux/backfile/Crux_v11n06_Jun.pdf"> Problem 938</a>, Crux Mathematicorum, page 199, Vol. 11, Jun. 1985.
%F A289980 3_n * (3_n + 1) = 1_n.2_n as 33 * 34 = 1122.
%F A289980 6_n * (6_n + 1)  = 4_n.2_n as 666 * 667 = 444222
%F A289980 9_n * (9_n + 1) = 9_n.0_n as 9999 * 10000 = 99990000.
%F A289980 G.f.: 2*x*(15000*x^14 + 22000*x^13 + 21000*x^12 - 1650*x^11 - 14420*x^10 - 17265*x^9 - 14865*x^8 - 8553*x^7 - 4374*x^6 + 1629*x^5 + 1082*x^4 + 645*x^3 - 15*x^2 - 10*x -6) / ((x-1) * (100*x^3-1) * (10*x^3-1) * (x^2+x+1)). - _Alois P. Heinz_, Jul 17 2017
%e A289980 6 * 7 = 42, 66 * 67 = 4422, 666 * 667 = 444222, 6666 * 6667 = 44442222.
%p A289980 seq(coeff(series(2*x*(15000*x^14+22000*x^13+21000*x^12-1650*x^11-14420*x^10-17265*x^9-14865*x^8-8553*x^7-4374*x^6+1629*x^5+1082*x^4+645*x^3-15*x^2-10*x-6)/((x-1)*(100*x^3-1)*(10*x^3-1)*(x^2+x+1)),x,n+1), x, n), n = 1 .. 30); # _Muniru A Asiru_, Dec 02 2018
%t A289980 Table[Select[Map[FromDigits[Join @@ Map[ConstantArray[#, n] &, #]] &, Drop[Tuples[Range[0, 9], {2}], 10]], IntegerQ@ Sqrt[4 # + 1] &], {n, 7}] // Flatten (* _Michael De Vlieger_, Jul 17 2017, after _Robert G. Wilson v_ at A002378 *)
%t A289980 CoefficientList[Series[2 (15000 x^14 + 22000 x^13 + 21000 x^12 - 1650 x^11 - 14420 x^10 - 17265 x^9 - 14865 x^8 - 8553 x^7 - 4374 x^6 + 1629 x^5 + 1082 x^4 + 645 x^3 - 15 x^2 - 10 x - 6) / ((x - 1) (100 x^3 - 1) (10 x^3 - 1) (x^2 + x + 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 02 2018 *)
%Y A289980 Cf. A002378, A181718, A276532.
%K A289980 nonn,easy,base
%O A289980 1,1
%A A289980 _Bernard Schott_, Jul 17 2017