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 A302537 #54 Jan 22 2025 06:01:40
%S A302537 1,8,16,25,35,46,58,71,85,100,116,133,151,170,190,211,233,256,280,305,
%T A302537 331,358,386,415,445,476,508,541,575,610,646,683,721,760,800,841,883,
%U A302537 926,970,1015,1061,1108,1156,1205,1255,1306,1358,1411,1465,1520,1576
%N A302537 a(n) = (n^2 + 13*n + 2)/2.
%C A302537 Binomial transform of [1, 7, 1, 0, 0, 0, ...].
%C A302537 Numbers m > 0 such that 8*m + 161 is a square.
%D A302537 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
%H A302537 G. C. Greubel, <a href="/A302537/b302537.txt">Table of n, a(n) for n = 0..5000</a>
%H A302537 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A302537 a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.
%F A302537 a(n) = a(n-1) + n + 6.
%F A302537 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.
%F A302537 a(n) = 2*a(n-1) - a(n-2) + 1.
%F A302537 a(n) = A004120(n+1) for n > 1.
%F A302537 a(n) = A056119(n) + 1.
%F A302537 a(n) = A152947(n+1) + A008589(n).
%F A302537 a(n) = A060544(n+1) - A002939(n).
%F A302537 a(n) = A000578(n+1) - A162261(n) for n > 0.
%F A302537 G.f.: (1 + 5*x - 5*x^2)/(1 - x)^3.
%F A302537 E.g.f.: (1/2)*(2 + 14*x + x^2)*exp(x).
%F A302537 Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - _Vaclav Kotesovec_, Apr 11 2018
%e A302537 Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
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%e A302537 ----------------------------------------------------------------------
%e A302537 . 1 8 16 25 35 46 58
%p A302537 a := n -> (n^2 + 13*n + 2)/2;
%p A302537 seq(a(n), n = 0 .. 100);
%t A302537 Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
%t A302537 CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
%t A302537 LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* _Robert G. Wilson v_, May 19 2018 *)
%o A302537 (Maxima) makelist((n^2 + 13*n + 2)/2, n, 0, 100);
%o A302537 (PARI) a(n) = (n^2 + 13*n + 2)/2; \\ _Altug Alkan_, Apr 12 2018
%o A302537 (Magma)
%o A302537 A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
%o A302537 [A302537(n): n in [0..50]]; // _G. C. Greubel_, Jan 21 2025
%o A302537 (Python)
%o A302537 def A302537(n): return (n**2 + 13*n + 2)//2
%o A302537 print([A302537(n) for n in range(51)]) # _G. C. Greubel_, Jan 21 2025
%Y A302537 Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
%Y A302537 A152947 (k = 0); A000124 (k = 1); A000217 (k = 2); A034856 (k = 3);
%Y A302537 A052905 (k = 4); A051936 (k = 5); A246172 (k = 6).
%Y A302537 Cf. A000578, A002939, A004120, A008589, A056119, A060544, A162261.
%K A302537 nonn,easy
%O A302537 0,2
%A A302537 _Franck Maminirina Ramaharo_, Apr 09 2018