OFFSET
1,1
COMMENTS
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.)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..300
Charles W. Trigg, Problem 938, Crux Mathematicorum, page 199, Vol. 11, Jun. 1985.
FORMULA
3_n * (3_n + 1) = 1_n.2_n as 33 * 34 = 1122.
6_n * (6_n + 1) = 4_n.2_n as 666 * 667 = 444222
9_n * (9_n + 1) = 9_n.0_n as 9999 * 10000 = 99990000.
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
EXAMPLE
6 * 7 = 42, 66 * 67 = 4422, 666 * 667 = 444222, 6666 * 6667 = 44442222.
MAPLE
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
MATHEMATICA
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 *)
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 *)
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Bernard Schott, Jul 17 2017
STATUS
approved