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A131941
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Partial sums of ceiling(n^2/2) (A000982).
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5
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0, 1, 3, 8, 16, 29, 47, 72, 104, 145, 195, 256, 328, 413, 511, 624, 752, 897, 1059, 1240, 1440, 1661, 1903, 2168, 2456, 2769, 3107, 3472, 3864, 4285, 4735, 5216, 5728, 6273, 6851, 7464, 8112, 8797, 9519, 10280, 11080, 11921, 12803, 13728, 14696, 15709
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Binomial transform of [0, 1, 1, 2, -2, 4, -8, 16, -32,...].
Starting with offset 1 = (1, 3, 5, 7,...) convolved with (1, 0, 3, 0, 5,...). - Gary W. Adamson, Feb 16 2009
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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FORMULA
| For even n a(n)=n*(2*n^2 +3*n +4)/12. For odd n a(n)=(n+1)*(2*n^2 +n +3)/12. - Washington Bomfim, Jul 31 2008
G.f.: x*(1+x^2)/(1+x)/(1-x)^4. a(n)= 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5). - R. J. Mathar, Feb 24 2010
Contribution from Mircea Merca, Oct 10 2010: (Start)
a(n) = round((2*n^3+3*n^2+4*n)/12) = round((2*n+1)*(2*n^2+3*n+3)/24) = floor((n+1)*(2*n^2+n+3)/12) = ceil((2*n^3+3*n^2+4*n)/12).
a(n) = a(n-2)+n^2-n+1, n>1. (End)
a(n) = (2*n*(2*n^2+3*n+4)-3*(-1)^n+3)/24. - Bruno Berselli, Dec 07 2010
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EXAMPLE
| a(3) = 8 = (0 + 1 + 2 + 5).
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MAPLE
| a(n):=round(1/(12)(2*n^(3)+3*n^(2)+4*n)) [Mircea Merca, Oct 10 2010]
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PROG
| (PARI) a(n) = (n+[0, 1][n%2+1]) * (2*n^2 +[3, 1][n%2+1]*n +[4, 3][n%2+1])/12 [Washington Bomfim, Jul 31 2008]
(MAGMA) [Round((2*n^3+3*n^2+4*n)/12): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
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CROSSREFS
| Cf. A000982.
Sequence in context: A122796 A104249 A025202 * A009858 A169947 A167616
Adjacent sequences: A131938 A131939 A131940 * A131942 A131943 A131944
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2007
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EXTENSIONS
| More terms from R. J. Mathar, Feb 24 2010
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