Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Dec 05 2016 05:27:45
%S 63,12285,2383290,462346038,89692748145,17399930794155,
%T 3375496881317988,654828995044895580,127033449541828424595,
%U 24643834382119669475913,4780776836681674049902590,927446062481862646011626610,179919755344644671652205659813
%N Numbers k such that 3*k+1 and 4*k+1 are both triangular numbers (A000217).
%H Colin Barker, <a href="/A279043/b279043.txt">Table of n, a(n) for n = 1..400</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3) for n>3.
%F G.f.: 63*x / ((1 - x)*(1 - 194*x + x^2)).
%e 63 is in the sequence because 3*63+1 = 190 and 4*63+1 = 253 are both triangular numbers.
%o (PARI) Vec(63*x / ((1 - x)*(1 - 194*x + x^2)) + O(x^20))
%o (PARI) isok(k) = ispolygonal(3*k+1, 3) & ispolygonal(4*k+1, 3)
%Y Cf. A000217, A274680.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Dec 04 2016