|
%I #17 Jun 29 2019 11:26:46
%S 0,1,100,8281,672400,54479161,4412944900,357449732641,28953439105600,
%T 21107054541321649,138483384602892402628,908589486379899193778809,
%U 5961255620138564686107812272,39111798123729126657669459066697,256612507489786800304910707633347364
%N Base-9 representation of a(n) is the concatenation of the base-9 representations of 1, 2, ..., n, n-1, ..., 1.
%C The base 9 is listed in A260343, because a(9) = A260851(9) = 21107054541321649 = 123456781087654321_9 is prime and therefore in A260852. See these sequences for more information.
%H D. Broadhurst, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;af419558.1508">Primes from concatenation: results and heuristics</a>, NmbrThry List, August 1, 2015
%F For n < b = 9, we have a(n) = A_b(n) = R(b,n)^2, where R(b,n) = (b^n-1)/(b-1) are the base-b repunits.
%e a(0) = 0 is the result of the empty sum corresponding to 0 digits.
%e a(2) = 100 = (9+1)^2 = 9^2 + 2*9 + 1 = 121_9, concatenation of (1, 2, 1).
%e a(10) = 1234567810111087654321_9 is the concatenation of (1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 10, 8, 7, 6, 5, 4, 3, 2, 1), where the middle "10, 11, 10" are the base-9 representations of 9, 10, 9.
%o (PARI) a(n,b=9)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))
%Y Base-9 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for the variants in other bases.
%K nonn,base
%O 0,3
%A _M. F. Hasler_, Aug 01 2015
|
