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 A269044 #97 May 31 2024 05:51:25
%S A269044 7,20,33,46,59,72,85,98,111,124,137,150,163,176,189,202,215,228,241,
%T A269044 254,267,280,293,306,319,332,345,358,371,384,397,410,423,436,449,462,
%U A269044 475,488,501,514,527,540,553,566,579,592,605,618,631,644,657,670,683,696,709,722,735
%N A269044 a(n) = 13*n + 7.
%C A269044 After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
%C A269044 This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
%C A269044 Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
%C A269044 The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
%C A269044 The primes of the sequence are listed in A140371.
%H A269044 Bruno Berselli, <a href="/A269044/b269044.txt">Table of n, a(n) for n = 0..1000</a>
%H A269044 Bruno Berselli, A description of the recursive method shown in the third comment: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian), 2008.
%H A269044 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A269044 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A269044 G.f.: (7 + 6*x)/(1 - x)^2.
%F A269044 a(n) = A088227(4*n+3).
%F A269044 a(n) = -A186113(-n-1).
%F A269044 Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
%F A269044 Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
%F A269044 E.g.f.: exp(x)*(7 + 13*x). - _Stefano Spezia_, Aug 02 2021
%t A269044 13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
%t A269044 LinearRecurrence[{2, -1}, {7, 20}, 60] (* _Vincenzo Librandi_, Feb 19 2016 *)
%o A269044 (Magma) [13*n+7: n in [0..60]];
%o A269044 (Maxima) makelist(13*n+7, n, 0, 60);
%o A269044 (PARI) vector(60, n, n--; 13*n+7)
%o A269044 (Sage) [13*n+7 for n in (0..60)]
%Y A269044 Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
%Y A269044 Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
%Y A269044 Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).
%K A269044 nonn,easy
%O A269044 0,1
%A A269044 _Bruno Berselli_, Feb 18 2016