A190090 Diagonal sums of the triangular matrix A190088.
1, 1, 4, 16, 42, 137, 443, 1365, 4316, 13625, 42785, 134758, 424331, 1335378, 4203927, 13233947, 41657808, 131135696, 412803240, 1299458257, 4090567673, 12876698159, 40534529294, 127598621869, 401667591501, 1264408966284, 3980231826575, 12529367967276, 39441185140197
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..201
- Index entries for linear recurrences with constant coefficients, signature (2,2,6,-3,0,1).
Programs
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Magma
[(&+[Binomial(3*n-4*k+1,3*n-6*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Mar 04 2018
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Mathematica
Table[Sum[Binomial[3n - 4k + 1, 3n - 6k + 1], {k, 0, n/2}], {n, 0, 26}] LinearRecurrence[{2,2,6,-3,0,1},{1,1,4,16,42,137},27] (* Harvey P. Dale, Jul 04 2011 *) -
Maxima
makelist(sum(binomial(3*n-4*k+1,3*n-6*k+1),k,0,n/2),n,0,12);
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PARI
Vec((1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6)+O(x^29)) \\ Charles R Greathouse IV, Jun 30 2011
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k+1,3*n-6*k+1).
G.f.: (1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6).
a(n) = 2*a(n-1)+ 2*a(n-2)+ 6*a(n-3)-3*a(n-4)+a(n-6), and a(0)=1, a(1)=1, a(2)=4, a(3)=16, a(4)=42, a(5)=137, . - Harvey P. Dale, Jul 04 2011