A165189 Partial sums of partial sums of (A001840 interleaved with zeros).
1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, 390, 462, 556, 650, 770, 890, 1040, 1190, 1375, 1560, 1785, 2010, 2280, 2550, 2871, 3192, 3570, 3948, 4389, 4830, 5341, 5852, 6440, 7028, 7700, 8372, 9136, 9900, 10764, 11628, 12600, 13572
Offset: 1
Keywords
Examples
A001840 interleaved with zeros is 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 9, 0, 12, 0, 15, 0, ... Partial sums thereof are 1, 1, 3, 3, 6, 6, 11, 11, 18, 18, 27, 27, 39, 39, 54, 54, ... This equals A014125 interleaved with itself. Partial sums thereof are 1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, ...
Links
- Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, 0, -2, -1, 4, -1, -2, 1).
Crossrefs
Programs
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Mathematica
Drop[Accumulate[Accumulate[Riffle[LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},30],0]]],2] (* or *) LinearRecurrence[{2,1,-4,1,2,0,-2,-1,4,-1,-2,1},{1,2,5,8,14,20,31,42,60,78,105,132},50] (* Harvey P. Dale, Jun 08 2018 *) -
PARI
/* first computes u = A001840 interleaved with zeros, then v = partial sums, then w = second partial sums */ {m=50; u=vector(m, n, polcoeff(x/((1-x^2)^3*(1+x^2+x^4))+x*O(x^(n)),n)); v=vector(m); a=u[1]; v[1]=a; for(n=2, m, a+=u[n]; v[n]=a); w=vector(m-1); a=v[1]; w[1]=a; for(n=2, m-1, a+=v[n]; w[n]=a); w} \\ Klaus Brockhaus, Sep 21 2009
Formula
G.f.: x/((1-x)^5*(1+x)^3*(1-x+x^2)*(1+x+x^2)).
54*a(n) = 631/64 +405/16*n +3/32*n^4 +15/8*n^3 +381/32*n^2 -(-1)^n*( 9/32*n^2 +45/16*n +375/64) -A131713(n) -3*A057079(n). - R. J. Mathar, Jun 16 2018
Extensions
Edited and corrected by R. J. Mathar, Klaus Brockhaus and N. J. A. Sloane, Sep 21 2009 - Sep 25 2009
Comments