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A267031
a(n) = (32*n^3 - 2*n)/3.
1
0, 10, 84, 286, 680, 1330, 2300, 3654, 5456, 7770, 10660, 14190, 18424, 23426, 29260, 35990, 43680, 52394, 62196, 73150, 85320, 98770, 113564, 129766, 147440, 166650, 187460, 209934, 234136, 260130, 287980, 317750, 349504, 383306, 419220, 457310, 497640, 540274, 585276, 632710, 682640, 735130, 790244
OFFSET
0,2
COMMENTS
This sequence alternates with the tetrahedral numbers, A000292, to create the centered octagonal pyramidal number sequence, A000447.
FORMULA
G.f.: 2*x*(5 + 22*x + 5*x^2)/(-1 + x)^4. - Michael De Vlieger, Jan 09 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 10 2016
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 9*log(2)/2 - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*(2-sqrt(2))*log(2)/4 - 3*sqrt(2)*log(sqrt(2)+2)/2. (End)
a(n) = binomial(4*n+1, 3). - Michel Marcus, Mar 05 2022
a(n) = 8*A000447(n) + A005843(n). - Yasser Arath Chavez Reyes, Mar 02 2024
EXAMPLE
a(4) = (32/3)*4^3 - (2/3)*4 = 680.
MATHEMATICA
Table[(32 n^3 - 2 n)/3, {n, 0, 42}] (* or *)
CoefficientList[Series[(2 x (5 + 22 x + 5 x^2))/(-1 + x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Jan 09 2016 *)
PROG
(Magma) [32/3*n^3-2/3*n: n in [0..35]]; // Vincenzo Librandi, Jan 10 2016
(PARI) concat(0, Vec(2*x*(5+22*x+5*x^2)/(1-x)^4 + O(x^100))) \\ Colin Barker, Jan 10 2016
(PARI) a(n) = (32*n^3 - 2*n)/3; \\ Altug Alkan, Jan 10 2015
CROSSREFS
Sequence in context: A226202 A271557 A351750 * A289163 A092718 A090763
KEYWORD
nonn,easy
AUTHOR
Peter M. Chema, Jan 09 2016
EXTENSIONS
More terms from Michael De Vlieger, Jan 09 2016
First term added from Vincenzo Librandi, Jan 10 2016
STATUS
approved