This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373028 #13 Jun 11 2024 10:07:58 %S A373028 2,3,4,5,6,1,8,9,10,2,12,13,14,3,16,10,18,4,20,21,2,5,24,25,10,6,28, %T A373028 29,30,7,32,33,34,8,36,3,38,9,20,41,42,10,44,45,46,11,48,49,50,12,4, %U A373028 53,54,13,56,33,58,14,60,23,62,15,64,65,66,5,68,69,70,17,72,73,74,18,76,49,78,19,80,33,6 %N A373028 a(n) is the least positive integer k such that 3*n^2 + 2*n + k^2 is a square. %C A373028 Many runs of consecutive integers appear in the sequence. The first ones are [2,3,4,5,6], [8,9,10], [12,13,14], [20,21], [24,25], [28,29,30], ... %C A373028 The scatterplot shows an interesting structure where distinct sets of terms can be seen on three straight lines. The greater the slope of the line, the higher the density of terms. The remaining terms are more randomly distributed between the three lines. %C A373028 More detailed observations: %C A373028 For the line of lowest slope, the term and term index parity alternates from one term to the next and if the term index is even, the term is odd and reciprocally. Terms of the same set and their indices are in arithmetic progression of respective common difference of 1 and 15. %C A373028 For the line of medium slope, all term indices are even and the term parity alternates from one term to the next. Terms of the same set and their indices are in arithmetic progression of respective common difference 1 and 4. %C A373028 For the line of greatest slope, if the term index is even, then the term is odd and reciprocally. All terms are equal to their index + 1. %C A373028 There are no fixed points a(n) = n since 3*n^2+2*n + n^2 = (2*n)^2 + 2*n falls between (2*n)^2 and (2*n+1)^2, so not square. %e A373028 a(1) = 2 because 3*1^2 + 2*1 = 5 and 5 + 1^2 is not a square, but 5 + 2^2 is. So, 2 is a term. %e A373028 a(2) = 3 because 3*2^2 + 2*2 = 16 and 16 + 1^2 and 16 + 2^2 are not squares,but 16 + 3^2 is. So, 3 is a term. %o A373028 (PARI) a(n) = my(m=3*n^2+2*n, k=1); while (!issquare(m+k^2), k++); k; \\ _Michel Marcus_, May 20 2024 %Y A373028 Cf. A129861, A373016, A373026. %K A373028 nonn,easy %O A373028 1,1 %A A373028 _Claude H. R. Dequatre_, May 20 2024