OFFSET
0,1
COMMENTS
Row sums of the triangle defined by non-interrupted runs in A080036.
If the sequence of integers is split at positions defined by A000124 we obtain A080036. Its runs of consecutive integers can be placed into rows of a triangle:
3;
5, 6;
8, 9, 10;
12, 13, 14, 15;
17, 18, 19, 20, 21;
...
The a(n) are the row sums of this triangle.
The a(n) are also the binomial transform of the quasi-finite sequence 3, 8, 8, 3, 0 (0 continued).
An associated integer sequence could be defined by a(n)/A026741(n+1) = 3, 11, 9, 27, ...
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = A162607(n+3) + n.
First differences: a(n+1) - a(n) = A104249(n+2), i.e., a(n) = a(n-1) + 3*n^2/2 + 7*n/2 +3.
Second differences: a(n+2) - 2*a(n+1) + a(n) = A016789(n+2).
a(n) = 2*a(n-1) - a(n-2) + 3*n + 5, n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3, n>2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
G.f.: (3-x+x^2)/(x-1)^4.
E.g.f.: (6 + 16*x + 8*x^2 + x^3)*exp(x)/2. - G. C. Greubel, Apr 21 2018
MATHEMATICA
Table[(6 + 10*n + 5*n^2 + n^3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {3, 11, 27, 54}, 50] (* G. C. Greubel, Apr 21 2018 *)
PROG
(Magma) [3+5*n+5*n^2/2+n^3/2: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011
(PARI) for(n=0, 50, print1((6+10*n+5*n^2+n^3)/2, ", ")) \\ G. C. Greubel, Apr 21 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Aug 28 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Aug 31 2009
Corrected typo in recurrence, observed by Paul Curtz - R. J. Mathar, Sep 25 2009
STATUS
approved