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 A349682 #54 Aug 31 2025 02:47:51
%S A349682 1,84,455,1330,2925,5456,9139,14190,20825,29260,39711,52394,67525,
%T A349682 85320,105995,129766,156849,187460,221815,260130,302621,349504,400995,
%U A349682 457310,518665,585276,657359,735130,818805,908600,1004731,1107414,1216865,1333300,1456935,1587986
%N A349682 a(n) = A000292(6*n + 1) where A000292 are the tetrahedral numbers.
%H A349682 Euclid of Alexandria, <a href="https://farside.ph.utexas.edu/books/Euclid/Elements.pdf">Elements</a>, VII Def. 17, p. 194, 300 BCE.
%H A349682 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A349682 a(n) = 1 + 11*n + 36*n^2 + 36*n^3 = (1 + 2*n)*(1 + 3*n)*(1 + 6*n).
%F A349682 G.f.: (1 + 80*x + 125*x^2 + 10*x^3)/(1 - x)^4. - _Stefano Spezia_, Nov 29 2021
%F A349682 From _Elmo R. Oliveira_, Aug 22 2025: (Start)
%F A349682 E.g.f.: exp(x)*(1 + 83*x + 144*x^2 + 36*x^3).
%F A349682 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%F A349682 From _Amiram Eldar_, Aug 31 2025: (Start)
%F A349682 Sum_{n>=0} 1/a(n) = Pi/(4*sqrt(3)) + 2*log(2) - 3*log(3)/4.
%F A349682 Sum_{n>=0} (-1)^n/a(n) = (3/4 - 1/sqrt(3))*Pi + sqrt(3)*log(2 + sqrt(3))/2 - log(2). (End)
%t A349682 nterms=50;Table[36n^3+36n^2+11n+1,{n,0,nterms-1}] (* _Paolo Xausa_, Nov 25 2021 *)
%o A349682 (PARI) a(n) = subst(m*(m+1)*(m+2)/6, 'm, 6*n+1); \\ _Michel Marcus_, Dec 16 2021
%o A349682 (Python)
%o A349682 def A349682(n): return n*(n*(36*n + 36) + 11) + 1 # _Chai Wah Wu_, Dec 27 2021
%Y A349682 Cf. A000292, A000447, A002492, A016921.
%K A349682 easy,nonn,changed
%O A349682 0,2
%A A349682 _Ralf Steiner_, Nov 25 2021