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%I #9 Jun 15 2021 01:46:11
%S 1,5,24,104,279,2001,4131,17453,88826,362532,1655660,6120642,25376649,
%T 128526482,301370205,1756488602,8046359747,30854867177,73845140753,
%U 488906501177,2106640948770,6573967883049,29711211505300
%N a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.
%C The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.
%H Antti Karttunen, <a href="/A179417/b179417.txt">Table of n, a(n) for n = 0..256</a>
%H Antti Karttunen, <a href="/A179417/a179417_256.png">Terms a(0)-a(255) drawn as a bit triangle, 1 pixel per bit.</a>
%H Antti Karttunen, <a href="/A179417/a179417_256_p2.png">Terms a(0)-a(255) drawn as a bit triangle, 2x2 pixels per bit.</a>
%H Antti Karttunen, <a href="/A179417/a179417_256_p3.png">Terms a(0)-a(255) drawn as a bit triangle, 3x3 pixels per bit.</a>
%e In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)=A165471(6)=A165471(7)=-1 and A165471(8)=A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.
%o (MIT Scheme)
%o (define (A179417 n) (let ((ul (A005408 n))) (let loop ((i (A000290 n)) (j 0) (s 0)) (cond ((= j ul) s) ((= 0 (1+halved (A165471 (1+ i)))) (loop (1+ i) (1+ j) s)) (else (loop (1+ i) (1+ j) (+ s (expt 2 j))))))))
%o (define (1+halved n) (floor->exact (/ (1+ n) 2)))
%Y Cf. A179418.
%Y Compare to similar bit triangle illustrations given in A080070, A122229, A122232, A122235, A122239, A122242, A122245.
%K nonn,base
%O 0,2
%A _Antti Karttunen_, Jul 27 2010
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