OFFSET
0,3
COMMENTS
Partial sums of A047209. - Reinhard Zumkeller, Jan 07 2012
From Wolfdieter Lang, Aug 06 2013: (Start)
a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*5)/4, the norm for integers a + b*omega(5), a, b rational integers, in the quadratic number field Q(sqrt(5)), where omega(5) = (1 + sqrt(5))/2 (golden section).
a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = n^2 + n*floor(n/2) - floor(n/2)^2 = floor(5*n^2/4) (the last eq. checks for even and odd n). (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = 5*n^2/4+((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
G.f.: x*(1+3*x+x^2)/(1-2*x+2*x^3-x^4). - Colin Barker, Jan 06 2012
a(n) = a(-n); a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>0, a(-1) = 1, a(0) = 0, a(1) = 1, a(2) = 5, n >= 3. (See the Bruno Berselli recurrence and a general comment for primes 1 (mod 4) under A227541). - Wolfdieter Lang, Aug 08 2013
a(n) = Sum_{j=1..n} Sum{i=1..n} ceiling((i+j-n+1)/2). - Wesley Ivan Hurt, Mar 12 2015
Sum_{n>=1} 1/a(n) = Pi^2/30 + tan(Pi/(2*sqrt(5)))*Pi/sqrt(5). - Amiram Eldar, Jan 16 2023
EXAMPLE
From Omar E. Pol, Sep 28 2011 (Start):
Illustration of initial terms (In a precise representation the pentagons should appear strictly concentric):
.
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o
. o o o o o o o o o o o o o
. o o o o o o o o
. o o o o o o o o o o o o o o o
.
. 1 5 11 20 31
.
(End)
MAPLE
MATHEMATICA
Table[Round[5n^2/4], {n, 0, 39}] (* Alonso del Arte, Sep 28 2011 *)
PROG
(PARI) a(n)=5*n^2>>2 \\ Charles R Greathouse IV, Sep 28 2011
(Magma) [5*n^2/4+((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011
(Haskell)
a032527 n = a032527_list !! n
a032527_list = scanl (+) 0 a047209_list
-- Reinhard Zumkeller, Jan 07 2012
(Python)
def A032527(n): return 5*n**2>>2 # Chai Wah Wu, Jul 30 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
New name from Omar E. Pol, Sep 28 2011
STATUS
approved