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A218329 Even 9-gonal (nonagonal) pyramidal numbers. 1

%I #14 Aug 01 2015 10:38:24

%S 10,34,80,266,420,624,1210,1606,2080,3290,4040,4896,6954,8170,9520,

%T 12650,14444,16400,20826,23310,25984,31930,35216,38720,46410,50610,

%U 55056,64714,69940,75440,87290,93654,100320,114586,122200,130144,147050,156026,165360

%N Even 9-gonal (nonagonal) pyramidal 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 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) = 3*a(n-3) - 3*a(n-6) + a(n-9) + 448.

%F a(n) = phi(n)*(phi(n)+9)*(7*phi(n)-36)/4374, where phi(n) = 3 + 12*n - 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3).

%F G.f.: 2*x*(5+12*x+23*x^2+78*x^3+41*x^4+33*x^5+29*x^6+3*x^7)/((1-x)^4*(1+x+x^2)^3).

%e The sequence of 9-gonal (nonagonal) pyramidal numbers A007584 begins 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210,.... As the third even term is 80, then a(3) = 80.

%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{10,34,80,266,420,624,1210,1606,2080,3290},39]

%Y Cf. A007584, A218328.

%K nonn

%O 1,1

%A _Ant King_, Oct 28 2012

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