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%I #9 Mar 01 2025 12:18:43
%S 11,0,2,27,92,352,1002,16927,2302,7827,25052,220052,13352,1487552,
%T 101752,195677,85177,137532552,173577
%N a(n) is the least nonnegative number that can be represented as the sum of two (not necessarily distinct) generalized pentagonal numbers in exactly n ways.
%C a(n) is the least k >= 0 such that A093518(k) = n.
%C a(17) > 5.4 * 10^7 if it exists.
%C From _Pontus von Brömssen_, Feb 28 2025: (Start)
%C a(19) > 3*10^9 if it exists.
%C After a(19), the following are all terms below 3*10^9:
%C n | a(n)
%C ---+-----------
%C 20 | 333802
%C 21 | 4891927
%C 22 | 391438802
%C 23 | 2543802
%C 24 | 494027
%C 25 | 55039427
%C 27 | 3764827
%C 28 | 8345052
%C 30 | 4339427
%C 32 | 2737177
%C 35 | 1375985677
%C 36 | 6422352
%C 38 | 429902552
%C 40 | 12350677
%C 41 | 85573502
%C 42 | 108485677
%C 45 | 94120677
%C 48 | 29014077
%C 50 | 733363177
%C 54 | 120983227
%C 56 | 308766927
%C 60 | 160558802
%C 63 | 2353016927
%C 64 | 101275552
%C 68 | 2139337552
%C 72 | 344336877
%C 80 | 725351927
%C 96 | 1073520852
%C (End)
%F A093518(a(n)) = n.
%e a(3) = 27 because 27 = 1 + 26 = 5 + 22 = 12 + 15 has 3 representations as the sum of two generalized pentagonal numbers, and no smaller number works.
%p GP:= [0,seq(op([m*(3*m-1)/2, m*(3*m+1)/2]),m=1..2000)]:
%p N:= GP[-1]:
%p V:= Array(0..N, datatype=integer[4]):
%p for i from 1 to nops(GP) do
%p for j from 1 to i do
%p r:= GP[i]+GP[j];
%p if r > N then break fi;
%p V[r]:= V[r]+1
%p od od:
%p W:= Array(0..16): count:= 0:
%p for i from 1 to N while count < 17 do
%p v:= V[i]; if v <= 16 and W[v] = 0 then W[v]:= i; count:= count + 1 fi
%p od:
%p W[1]:= 0:
%p convert(W,list);
%Y Cf. A001318, A093518.
%K nonn,more
%O 0,1
%A _Robert Israel_, Feb 26 2025
%E a(17)-a(18) from _Pontus von Brömssen_, Feb 28 2025