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 A336535 #50 Jul 13 2025 17:20:31
%S A336535 1,6,28,91,231,496,946,1653,2701,4186,6216,8911,12403,16836,22366,
%T A336535 29161,37401,47278,58996,72771,88831,107416,128778,153181,180901,
%U A336535 212226,247456,286903,330891,379756,433846,493521,559153,631126,709836,795691,889111,990528,1100386,1219141,1347261,1485226,1633528,1792671
%N A336535 a(n) = (m(n)^2 + 3)*(m(n)^2 + 7)/32, where m(n) = 2*n - 1.
%C A336535 For m(n) = 3,5,11, and 181, the perfect numbers (A000396), 6, 28, 496, and 33550336 are produced, respectively. 3,5,11, and 181 are the numbers m(n) such that (m(n)^2+7) is a power of 2. cf A038198.
%C A336535 The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A000217(n). - _Miguel-Ángel Pérez García-Ortega_, May 16 2025
%D A336535 David M. Burton, Elementary Number Theory, McGraw-Hill (2011), 25.
%H A336535 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A336535 From _Stefano Spezia_, Jul 25 2020: (Start)
%F A336535 O.g.f.: x*(1 + x + 8*x^2 + x^3 + x^4)/(1 - x)^5.
%F A336535 a(n) = (1 - n + n^2)*(2 - n + n^2)/2.
%F A336535 a(n) = A002061(n)*A014206(n-1)/2.
%F A336535 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
%F A336535 a(n) = (A000217(n-1)+1)*(2*A000217(n-1)+1). - _Miguel-Ángel Pérez García-Ortega_, May 16 2025
%e A336535 m(2) = 2*2-1 = 3 and (3^2+3)*(3^2+7)/32 = 6, so 6 is in the sequence.
%t A336535 Table[((n^2+3)*(n^2+7))/32, {n,1,100,2}]
%t A336535 a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* _Miguel-Ángel Pérez García-Ortega_, May 16 2025 *)
%o A336535 (PARI) a(n)=(1-n+n^2)*(2-n+n^2)/2 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A336535 Cf. A000396, A038198.
%Y A336535 Cf. A002061, A014206.
%Y A336535 Cf. A000217, A058919, A383833, A383834.
%K A336535 nonn,easy
%O A336535 1,2
%A A336535 _Jeff Brown_, Jul 24 2020