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A204539 a(n) is the number of integers N=4k whose "basin" sequence (cf. comment) ends in n^2.

%I A204539 #56 Mar 30 2025 14:16:10
%S A204539 1,1,1,2,1,3,2,4,2,4,3,5,1,9,2,10,3,5,7,9,2,10,9,9,2,13,9,8,4,20,4,15,
%T A204539 6,15,8,12,6,22,6,15,15,21,5,13,12,23,7,24,11,19,15,24,6,30,6,26,7,27,
%U A204539 26,13,6,33,27,30,5,13,30,30,5,37,15,26,28,32,7,17,25,54,9,30,21,41,25
%N A204539 a(n) is the number of integers N=4k whose "basin" sequence (cf. comment) ends in n^2.
%C A204539 The "basin" (analogous to river basins, for reasons set out below) is the number of positive integers N=4k which end in the "sea" at n^2. The "sea" of N is found as follows:
%C A204539 Starting out with N, in step i=1,2,3,..., stop if you have reached N=(i+1)^2 (the "sea" of N), otherwise set N to the next higher, odd or even (according to the parity of i), multiple of i+2, and go to step i+1.
%C A204539 Partial sums of this sequence appear to be A104738 (with a shift in offset). This has been confirmed for at least the first 4000 terms, but it is not at all clear why this is the case. - _Ray Chandler_, Jan 20 2012
%C A204539 After the first term, this sequence agrees with A028914 except for offset. Therefore this sequence is related to A028913, A007952, A002491 and A108696 dealing with the sieve of Tchoukaillon (or Mancala, or Kalahari). - _Ray Chandler_, Jan 20 2012
%H A204539 Ray Chandler, <a href="/A204539/b204539.txt">Table of n, a(n) for n = 2..10001</a>
%H A204539 Mark Dukes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Dukes/dukes3.html">Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
%H A204539 Mark Dukes, <a href="https://arxiv.org/abs/2202.02381">Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire</a>, arXiv:2202.02381 [math.NT], 2022.
%e A204539 For integers N=4,8,12,16,... we have the following sequences:
%e A204539   {4}
%e A204539   {8, 9}   (8 -> the next higher odd multiple of 3, which is 9 -> STOP)
%e A204539   {12, 15, 16}  (12 -> 3*5=15 -> 4*4=16 -> STOP)
%e A204539   {16, 21, 24, 25}
%e A204539   {20, 21, 24, 25}
%e A204539   {24, 27, 32, 35, 36}
%e A204539   {28, 33, 40, 45, 48, 49}
%e A204539   {32, 33, 40, 45, 48, 49}
%e A204539   {36, 39, 40, 45, 48, 49}
%e A204539   ...
%e A204539 Thus there is 1 integer N=4k ending in the sea at 2^2, whence basin a(2)=1, and idem for 3 and 4.
%e A204539 The two integers 16 and 20 end at 5^2, so the basin of 5 is a(5)=2.
%e A204539 There is again a(6)=1 integer ending in 6^2, while the basin of 7 are the 3 integers 28, 32, and 36, which all merge into the "river" that enters the "sea" in 7^2=49.
%e A204539 Thus the first 6 terms in the sequence are 1, 1, 1, 2, 1, 3.
%e A204539 Take N=100 as an example: the next integer on the same line is the next higher odd multiple of 3, i.e., smallest 3*(2m+1) > 100, which is 105. The next number is the least even multiple of 4, 4*(2m) = 112, etc., leading to 115 = 5*(2m+1), followed by 120 = 6*(2m), 133 = 7*(2m+1), 144 = 8*2m (where we have a square, but not the square of 8), 153 =9*(2m+1), 160 = 10*2m, 165 = 11*(2m+1), 168 = 12*(2m) and finally 169 = 13*13.
%t A204539 cumul[n_Integer] := Module[{den1 = n, num = n^2, den2}, While[num > 4 && den1 != 2, num = num - 1; den1 = den1 - 1; den2 = Floor[num/den1]; If[Not[EvenQ[den1 + den2]], den2 = den2 - 1]; num = den1 den2]; Return[num/4]]; basin[2] := 1; basin[n_Integer] := cumul[n] - cumul[n - 1]; Table[basin[n], {n, 2, 75}] (* _Alonso del Arte_, Jan 19 2012 *)
%o A204539 (PARI) bs(n,s,m=2)={while(n>m^2,n=(n\m+++2-bittest(n\m-m,0))*m; s & print1(n","));n}
%o A204539 n=4; for(c=2,50, for(k=1,9e9, bs(n+=4)==c^2 || print1(k",")||break)) \\ _M. F. Hasler_, Jan 20 2012
%Y A204539 Cf. A002491, A007952, A028913, A104738, A108696, A204540.
%Y A204539 Essentially the same as A028914.
%K A204539 nonn
%O A204539 2,4
%A A204539 _Colm Fagan_, Jan 16 2012