|
|
A165189
|
|
Partial sums of partial sums of (A001840 interleaved with zeros).
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Also convolution of period six sequence 1,0,0,0,0,0,1,... (A079979) with sequence 1,2,5,8,14,20,30,40,... (A006918 without initial zero).
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, 0, -2, -1, 4, -1, -2, 1).
|
|
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
|
|
EXAMPLE
|
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, ...
|
|
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 *)
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|