OFFSET
0,4
COMMENTS
Partial sums of A037597.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..170
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (5,-4,1,-5,4).
FORMULA
a(n) = round((8*4^n - 24*n - 17)/54).
a(n) = floor(4*(4^n - 3*n - 1)/27).
a(n) = ceiling((4*4^n - 12*n - 13)/27).
a(n) = round(4*(4^n - 3*n - 1)/27).
a(n) = a(n-3) + (7*4^(n-2) - 4)/3 , n > 3.
a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5), n > 5.
G.f.: x^2*(1+3*x) / ( (1-4*x)*(1+x+x^2)*(1-x)^2 ).
EXAMPLE
a(4) = 0 + 1 + 7 + 28 = 36.
MAPLE
A178744 := proc(n) add( floor(4^i/9), i=0..n) ; end proc:
MATHEMATICA
Table[Floor[4*(4^n-3*n-1)/27], {n, 0, 30}] (* G. C. Greubel, Jan 24 2019 *)
Accumulate[Floor[4^Range[0, 30]/9]] (* or *) LinearRecurrence[{5, -4, 1, -5, 4}, {0, 0, 1, 8, 36}, 30] (* Harvey P. Dale, Jul 13 2024 *)
PROG
(Magma) [&+[Floor(4^k/9): k in [0..n]]: n in [0..25]]; // Bruno Berselli, Apr 26 2011
(PARI) vector(30, n, n--; (4*(4^n-3*n-1)/27)\1) \\ G. C. Greubel, Jan 24 2019
(SageMath) [floor(4*(4^n-3*n-1)/27) for n in (0..30)] # G. C. Greubel, Jan 24 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Mircea Merca, Dec 26 2010
STATUS
approved