%I #28 May 10 2019 14:42:11
%S 52029,316725,1093345,2811129,6031029,11445709,19879545,32288625,
%T 49760749,73515429,104903889,145409065,196645605,260359869,338429929,
%U 432865569,545808285,679531285,836439489,1019069529,1230089749,1472300205,1748632665,2062150609
%N An infinite sequence of 4-digit half-palindromes.
%C For the definition of k-digit half-palindromes, see A191279. Although there exist infinitely many polynomials taking only 3-digit half-palindrome values (see comment to A191279), only two polynomials are known with all values 4-digit half-palindromes. They are the polynomials P(n) which were discovered in SeqFan Discussion list from Mar 14 2011 and its double.
%C The sequence lists values of P(n). All these are odd. For a given k>=5, up to now it is unknown if there are polynomials taking only k-digit half-palindrome values and it is unknown whether there exist infinitely many such numbers.
%C Conjecture. For every k>=2, there exists a polynomial of degree k taking only k-digit half-palindrome values.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (2*n+2)(14*n+9)^3+(3*n+3)*(14*n+9)^2+(5*n+3)*(14*n+9)+(2*n+1) = (2*n+1)*(14*n+11)^3 +(5*n+3)*(14*n+11)^2 +(3*n+3)*(14*n+11) +(2*n+2), such that the bases b < c are b=14*n+9, c=14*n+11.
%F G.f. -x*(52029+56580*x+30010*x^2-8636*x^3+1729*x^4) / (x-1)^5. - _R. J. Mathar_, Jul 01 2012
%t LinearRecurrence[{5,-10,10,-5,1},{52029,316725,1093345,2811129,6031029},20] (* _Harvey P. Dale_, Sep 19 2018 *)
%Y Cf. A002113, A006995, A059809, A191279.
%K nonn,easy,base
%O 1,1
%A _Vladimir Shevelev_, May 30 2011