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A015219 Odd tetrahedral numbers: a(n) = (4*n+1)*(4*n+2)*(4*n+3)/6.

%I A015219 #57 Aug 15 2025 20:41:51
%S A015219 1,35,165,455,969,1771,2925,4495,6545,9139,12341,16215,20825,26235,
%T A015219 32509,39711,47905,57155,67525,79079,91881,105995,121485,138415,
%U A015219 156849,176851,198485,221815,246905,273819,302621,333375,366145,400995,437989,477191,518665
%N A015219 Odd tetrahedral numbers: a(n) = (4*n+1)*(4*n+2)*(4*n+3)/6.
%H A015219 Delbert L. Johnson, <a href="/A015219/b015219.txt">Table of n, a(n) for n = 0..19999</a>
%H A015219 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A015219 From _Jaume Oliver Lafont_, Oct 20 2009: (Start)
%F A015219 G.f.: (1+x)*(1+30*x+x^2)/(1-x)^4.
%F A015219 Sum_{n>=0} 1/a(n) = (3/2)*log(2). (End)
%F A015219 From _Ant King_, Oct 19 2012: (Start)
%F A015219 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A015219 a(n) = 64 + 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A015219 a(n) = A000292(4*n+1). - _L. Edson Jeffery_, Jan 16 2013
%F A015219 a(n) = A000447(2*n+1). - _Michel Marcus_, Jan 25 2016
%F A015219 Sum_{n>=0} (-1)^n/a(n) = 3*(sqrt(2)-1)*Pi/4. - _Amiram Eldar_, Jan 04 2022
%F A015219 a(n) = A001505(n)/6. - _R. J. Mathar_, Apr 17 2024
%F A015219 E.g.f.: exp(x)*(3 + 102*x + 144*x^2 + 32*x^3)/3. - _Elmo R. Oliveira_, Aug 15 2025
%t A015219 LinearRecurrence[{4, -6, 4, -1}, {1, 35, 165, 455}, 35] (* _Ant King_, Oct 19 2012 *)
%t A015219 Table[(4 n + 1) (4 n + 2) (4 n + 3)/6, {n, 0, 40}] (* _Vincenzo Librandi_, Jan 25 2016 *)
%o A015219 (PARI) a(n)=binomial(4*n+3,3) \\ _Charles R Greathouse IV_, Jan 16 2013
%o A015219 (Magma) [(4*n+1)*(4*n+2)*(4*n+3)/6: n in [0..40]]; // _Vincenzo Librandi_, Jan 25 2016
%Y A015219 Cf. A000292, A000447, A001505.
%K A015219 nonn,easy
%O A015219 0,2
%A A015219 _Mohammad K. Azarian_
%E A015219 More terms from _Erich Friedman_