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approved
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Revision History for A005232
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PROG
(PARI) {a(n) = (n^3 + 9*n^2 + (32-9*(n%2))*n + [48, 15, 36, 15][n%4+1]) / 48}; /* _\\ _Michael Somos_, Feb 01 2007 */
(PARI) {a(n) = my(s=1); if( n<-5, n = -6-n; s=-1); if( n<0, 0, s * polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; /* _\\ _Michael Somos_, Feb 01 2007 */
(pythonPython) a=lambda n: sum((k//2+1)*((n-k)//2+1) for k in range((n-1)//2+1))+(n+1)%2*(((n//4+1)*(n//4+2))//2) # Gabriel Burns, Nov 08 2016
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editing
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proposed
approved
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editing
proposed
FORMULA
G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)) . - Vladeta Jovovic, Aug 05 2000
Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1 ]. - Michael Somos, Feb 01 2007
a(2n+1) = A006918(2n+2)/2. a(2n) = (A006918(2n+1) + A008619(n))/2.;
a(2n) = (A006918(2n+1) + A008619(n))/2.
if n == 0 (mod 4, ), then a(n) = n*(n^2-3*n+8)/48;
if n == 1, 3 (mod 4, ), then a(n) = (n^2-1)*(n-3)/48;
if n == 2 (mod 4, ), then a(n) = (n-2)*(n^2-n+6)/48. (End)
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-7) - a(n-8) with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 8, a(5) = 10, a(6) = 16, a(7) = 20. - Harvey P. Dale, Oct 24 2012
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approved
editing
LINKS
Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
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reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed