OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).
FORMULA
a(n) = floor(4^(n+1)/15) = 4^(n+1)/15 - 1/6 - (-1)^n/10. - Benoit Cloitre, Apr 18 2003
G.f.: 1/((1-x)*(1+x)*(1-4*x)); a(n) = 3*a(n-1) + 4*a(n-2)+1. Partial sum of A015521. - Paul Barry, Nov 12 2003
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k); a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(j+k)*4^j. - Paul Barry, Nov 12 2003
Convolution of A000302 and A059841 (4^n and periodic{1, 0}). a(n) = Sum_{k=0..n} (1 + (-1)^(n-k))*4^k/2. - Paul Barry, Jul 19 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*(J(2*k+1)-1)/2, J(n)=A001045(n). - Paul Barry, Mar 06 2008
a(n) = round((8*4^n-5)/30) = ceiling((4*4^n-4)/15) = round((4*4^n-4)/15); a(n) = a(n-2) + 4^(n-1), n > 1. - Mircea Merca, Dec 28 2010
a(n) = A117616(n)/2. - J. M. Bergot, Apr 22 2015
a(n)+a(n+1) = A002450(n+1). - R. J. Mathar, Feb 27 2019
MAPLE
seq(floor((4^(n+1)-1)/15), n=1..25) # Mircea Merca, Dec 28 2010
MATHEMATICA
Join[{a=1, b=4}, Table[c=3*b+4*a+1; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
PROG
(Magma) [Round((8*4^n-5)/30): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
STATUS
approved