A015219 Odd tetrahedral numbers: a(n) = (4*n+1)*(4*n+2)*(4*n+3)/6.
1, 35, 165, 455, 969, 1771, 2925, 4495, 6545, 9139, 12341, 16215, 20825, 26235, 32509, 39711, 47905, 57155, 67525, 79079, 91881, 105995, 121485, 138415, 156849, 176851, 198485, 221815, 246905, 273819, 302621, 333375, 366145, 400995, 437989, 477191, 518665
Offset: 0
Links
- Delbert L. Johnson, Table of n, a(n) for n = 0..19999
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[(4*n+1)*(4*n+2)*(4*n+3)/6: n in [0..40]]; // Vincenzo Librandi, Jan 25 2016
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Mathematica
LinearRecurrence[{4, -6, 4, -1}, {1, 35, 165, 455}, 35] (* Ant King, Oct 19 2012 *) Table[(4 n + 1) (4 n + 2) (4 n + 3)/6, {n, 0, 40}] (* Vincenzo Librandi, Jan 25 2016 *) -
PARI
a(n)=binomial(4*n+3,3) \\ Charles R Greathouse IV, Jan 16 2013
Formula
From Jaume Oliver Lafont, Oct 20 2009: (Start)
G.f.: (1+x)*(1+30*x+x^2)/(1-x)^4.
Sum_{n>=0} 1/a(n) = (3/2)*log(2). (End)
From Ant King, Oct 19 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 64 + 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A000292(4*n+1). - L. Edson Jeffery, Jan 16 2013
a(n) = A000447(2*n+1). - Michel Marcus, Jan 25 2016
Sum_{n>=0} (-1)^n/a(n) = 3*(sqrt(2)-1)*Pi/4. - Amiram Eldar, Jan 04 2022
a(n) = A001505(n)/6. - R. J. Mathar, Apr 17 2024
E.g.f.: exp(x)*(3 + 102*x + 144*x^2 + 32*x^3)/3. - Elmo R. Oliveira, Aug 15 2025
Extensions
More terms from Erich Friedman