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A019298
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Balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).
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13
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0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186
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OFFSET
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0,3
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COMMENTS
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Alternately add and subtract successively longer sets of integers: 0; 1=0+1; -4=1-2-3; 11=-4+4+5+6; -23=11-7-8-9-10; 42=-23+11+12+13+14+15; -69=42-16-17-18-19-20-21; ... then take absolute values. - Walter G. Carlini (541carlini(AT)charter.net), Aug 28 2003
Number of 3 X 3 symmetric matrices with nonnegative integer entries, such that every row (and column) sum equals n-1.
Equals sum_{0..n} of "three-quarter squares" sequence (A077043) - Philipp M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007
a(n) = sum of n-th row in A220075, n > 0. - Reinhard Zumkeller, Dec 03 2012
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_3(lambda).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.14(a), p. 452.
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LINKS
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Table of n, a(n) for n=0..37.
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 13.
Index to sequences with linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).
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FORMULA
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a(n) = floor((n^2+1)(2n+3)/8). G.f.: x(x^2+x+1)/((x+1)(x-1)^4).
a(n) = floor((2n^3 + 3n^2 + 2n)/8); also nearest integer to ((n+1)^4 - n^4)/16.
a(n) = (4n^3+6n^2+4n+1-(-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Mar 06 2004
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MAPLE
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series(x*(x^2+x+1)/(x+1)/(x-1)^4, x, 80);
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MATHEMATICA
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Table[ Ceiling[3*n^2/4], {n, 0, 37}] // Accumulate (* Jean-François Alcover, Dec 20 2012, after Philipp M. Buluschek's comment *)
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PROG
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(PARI) a(n)=(n^2+1)*(2*n+3)\8 \\ Charles R Greathouse IV, Apr 04 2013
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CROSSREFS
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Cf. A053493, A077043 (first differences), A002717.
Sequence in context: A027378 A092498 A131177 * A173702 A014242 A008181
Adjacent sequences: A019295 A019296 A019297 * A019299 A019300 A019301
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Eric E Blom (eblom(AT)REM.re.uokhsc.edu)
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EXTENSIONS
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Error in n=8 term corrected May 15 1997
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STATUS
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approved
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