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Numbers k such that k^2 = (2i+1)^2 + (2j+1)^2 - 1, where i <= j and i,j > 0, retaining multiples of same k obtained from different i, j combinations.
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%I #33 Feb 18 2019 02:07:55

%S 7,13,17,21,23,27,31,33,37,41,43,43,47,47,47,53,55,57,57,57,63,67,73,

%T 73,73,75,77,81,83,83,83,87,89,91,93,93,97,99,99,103,105,107,107,109,

%U 111,113,115,117,119,123,123,123,125,127,129,133,133,133,135,137,143,143,147,149

%N Numbers k such that k^2 = (2i+1)^2 + (2j+1)^2 - 1, where i <= j and i,j > 0, retaining multiples of same k obtained from different i, j combinations.

%C At i = 0 and at j = 0 solutions include all odd numbers, so these have not been listed above, but the full table (for all i,j) is considered for patterns later below.

%C If we exclude multiples of the same k in the listed sequence it equals A180263, which uses a different set of criteria related to nonprimes.

%C The results may be viewed as a "distance" from a point on the real x-y plane to the imaginary point sqrt(-1) on a complex z-axis (or plane).

%C These results occur in a number of infinite and branching "subsequences" with distinct but usually simple patterns that are discernible when viewing the triangle. The main subsequence (#1) begins at value 3. Other subsequences branch off from it. Patterns are evident in both the values of results (by examining differences) and in their i and j positions. Example subsequences are numbered in the order of lowest value for their starting terms:

%C Subseq #1: 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, etc.

%C Subseq #2: 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, etc.

%C Subseq #3: 7, 41, 75, 99, 133, 157, 191, 215, 249, 273, 307, etc.

%C Subseq #4: 13, 21, 47, 55, 81, 89, 115, 123, 149, 157, 183, 191, etc.

%C Subseq #5: 17, 23, 47, 57, 93, 107, 155, 173, 233, 255, 327, 353, etc.

%C Subseq #6: 21, 31, 47, 57, 73, 83, 99, 109, 125, 135, 151, 161, etc.

%C Subsequence #1 conforms to A002061 (when excluding its two initial 1's) with the 3 at i=1, j=0 (on other side of diagonal). Subsequence #2 conforms to A063226, when excluding its initial term 3. The other examples given are not in the OEIS currently.

%C Branching can be seen as follows: subsequence #3 starts on the diagonal at i,j = 2 and it branches off #1 (or #2). Subsequence #4 branches off #1. Subsequence #5 branches off of #2. Subsequence #6 branches off #1. Subsequence #5 and #6 cross over each other, at (47,57) which occurs at the same i,j, values.

%C The full set of odd numbers along the axes for i,j = 0, in all four quadrants are also simple patterns.

%C Additional subsequences are spawned at higher values of i and j, apparently without end, evoking the appearance of linear and curved fractals. It is NOT clear whether all instances of k occur in such infinite subsequences.

%C Results are prolific because of the -1 in the equation. Without it (i.e., using only Pythagorean distance) there are no integer solutions for k with two odd legs to the triangle. Though certain other constants (added or subtracted) produce multiple obvious subsequences as well.

%e 7 is a term because 7^2 = 5^2 + 5^2 - 1.

%e 13 is a term because 13^2 = 7^2 + 11^2 - 1.

%K nonn

%O 1,1

%A _Richard R. Forberg_, May 27 2013