A243099 A002061 and A000217 interleaved.
0, 0, 0, 1, 1, 3, 3, 7, 6, 13, 10, 21, 15, 31, 21, 43, 28, 57, 36, 73, 45, 91, 55, 111, 66, 133, 78, 157, 91, 183, 105, 211, 120, 241, 136, 273, 153, 307, 171, 343, 190, 381, 210, 421, 231, 463, 253, 507, 276, 553, 300, 601
Offset: 3
Links
- Paolo Xausa, Table of n, a(n) for n = 3..10000
- Kival Ngaokrajang, Illustration of initial terms.
- Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).
Crossrefs
Programs
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Mathematica
With[{r=Range[50]}, Join[{0, 0, 0}, Riffle[r^2-r+1, PolygonalNumber[r]]]] (* or *) LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 0, 0, 1, 1, 3, 3, 7}, 100] (* Paolo Xausa, Feb 09 2024 *) -
PARI
a(n) = if(n<6,0,if(Mod(n,2)==0,(n/2-2)^2-(n/2-2)+1,(n/2-5/2)*(n/2-5/2+1)/2)) for (n=3,100,print1(a(n),", "))
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PARI
concat([0,0,0], Vec(-x^6*(x^4+x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ Colin Barker, Aug 19 2014
Formula
a(3) = a(4) = a(5) = 0; for n >= 6, a(n) = (n/2-2)^2-(n/2-2)+1 if even n, a(n) = (n/2-5/2)*(n/2-5/2+1)/2 if odd n.
From Colin Barker, Aug 19 2014: (Start)
a(n) = (71+41*(-1)^n-4*(7+3*(-1)^n)*n+(3+(-1)^n)*n^2)/16 for n>4.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>10.
G.f.: -x^6*(x^4+x+1) / ((x-1)^3*(x+1)^3). (End)
Sum_{n>=6} (-1)^(n+1)/a(n) = 2 - tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Feb 11 2024
Comments