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Even tetrahedral numbers.
1

%I #37 Mar 07 2022 13:05:43

%S 0,4,10,20,56,84,120,220,286,364,560,680,816,1140,1330,1540,2024,2300,

%T 2600,3276,3654,4060,4960,5456,5984,7140,7770,8436,9880,10660,11480,

%U 13244,14190,15180,17296,18424,19600,22100,23426,24804,27720,29260,30856,34220,35990

%N Even tetrahedral numbers.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3,0,-3,3,0,1,-1).

%F From _Ant King_, Oct 19 2012: (Start)

%F a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10).

%F a(n) = 64 + 3*a(n-3) - 3*a(n-6) + a(n-9).

%F G.f.: 2*x*(2+3*x+5*x^2+12*x^3+5*x^4+3*x^5+2*x^6) / ((1-x)^4*(1+x+x^2)^3).

%F Sum_{n>=1} 1/a(n) = 3/2*(1-log(2)). (End)

%F From _Amiram Eldar_, Mar 07 2022: (Start)

%F a(n) = A000292(A004772(n+1)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) - 15/2 + 9*sqrt(2)*log(sqrt(2)+1)/2. (End)

%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{0,4,10,20,56,84,120,220,286,364},41] (* _Ant King_, Oct 19 2012 *)

%t Select[Table[(Times@@(n+{0,1,2}))/6,{n,0,60}],EvenQ] (* _Harvey P. Dale_, Jan 22 2013 *)

%Y Cf. A000292, A004772, A015219.

%K nonn,easy

%O 0,2

%A _Mohammad K. Azarian_

%E More terms from _Erich Friedman_

%E a(0) prepended by _Amiram Eldar_, Mar 07 2022