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A113439
a(n) = a(n-1) + Sum_{k=1..floor(n/4)} a(n-4k), with a(0)=1.
7
1, 1, 1, 1, 2, 3, 4, 5, 8, 12, 17, 23, 34, 50, 72, 101, 146, 212, 306, 436, 627, 905, 1305, 1871, 2689, 3872, 5577, 8014, 11521, 16576, 23858, 34309, 49337, 70968, 102108, 146868, 211233, 303832, 437080, 628708, 904306, 1300737, 1871065, 2691401
OFFSET
0,5
COMMENTS
If presented in four rows a(4k), a(4k+1), a(4k+2) and a(4k+3), each term is the sum of the previous term in the sequence and the partial sum of its row; see Example section.
FORMULA
a(n) = a(n-1) + 2*a(n-4) - a(n-5).
a(n) = 9*a(n-4) - 28*a(n-8) + 38*a(n-12) - 20*a(n-16) +a(n-20).
G.f.: (1-x^4)/(1-x-2*x^4+x^5).
EXAMPLE
From Jon E. Schoenfield, Mar 11 2017: (Start)
Table of values T(j,k) = a(4k+j) in 4 rows:
.
j | k=0 1 2 3 4 5 6 7
----+--------------------------------------------------
0 | 1 2 8 34 146 627 2689 11521 ...
1 | 1 3 12 50 212 905 3872 16576 ...
2 | 1 4 17 72 306 1305 5577 23858 ...
3 | 1 5 23 101 436 1871 8014 34309 ...
.
T(2,4) = T(1,4) + T(2,0) + T(2,1) + T(2,2) + T(2,3)
306 = 212 + 1 + 4 + 17 + 72
(End)
MATHEMATICA
CoefficientList[Series[(1 - x^4)/(1 - x - 2*x^4 + x^5), {x, 0, 50}], x] (* G. C. Greubel, Mar 11 2017 *)
LinearRecurrence[{1, 0, 0, 2, -1}, {1, 1, 1, 1, 2}, 50] (* Harvey P. Dale, Nov 10 2019 *)
PROG
(PARI) x='x+O('x^50); Vec((1-x^4)/(1-x-2*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Floor van Lamoen, Nov 04 2005
STATUS
approved