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A002417
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4-dimensional figurate numbers: n*C(n+2,3).
(Formerly M4506 N1907)
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92
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1, 8, 30, 80, 175, 336, 588, 960, 1485, 2200, 3146, 4368, 5915, 7840, 10200, 13056, 16473, 20520, 25270, 30800, 37191, 44528, 52900, 62400, 73125, 85176, 98658, 113680, 130355, 148800, 169136, 191488, 215985, 242760, 271950, 303696, 338143, 375440, 415740
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 colors. - R. H. Hardin (rhhardin(AT)att.net), Feb 23 2002
a(n) is the sum of all numbers that cannot be written as t*(n+1) + u*(n+2) for nonnegative integers t,u (see Schuh). - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 09 2002
a(n) is the total number of rectangles (including squares) contained in a stepped pyramid of n rows (or of base 2n-1) of squares. A stepped pyramid of squares of base 2*6 - 1 = 11, for instance, has the following vertices:
..........X.X
........X.X.X.X
......X.X.X.X.X.X
....X.X.X.X.X.X.X.X
..X.X.X.X.X.X.X.X.X.X
X.X.X.X.X.X.X.X.X.X.X.X
X.X.X.X.X.X.X.X.X.X.X.X - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 02 2003
a(n) equals -1 times the coefficient of x^3 of the characteristic polynomial of the (n + 2) X (n + 2) matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). [From John M. Campbell, May 28, 2011]
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
K. -W. Lau, Solution to Problem 2495, Journal of Recreational Mathematics 2002-3 31(1) 79-80.
Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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FORMULA
| a(n) = n^2*(n+1)*(n+2)/6.
G.f.: x*(1+3*x)/(1-x)^5.
a(n) = C(n+2, 2)*n^2/3 - Paul Barry (pbarry(AT)wit.ie), Jun 26 2003
a(n) = C(n+3, n)*C(n+1, 1) - Zerinvary Lajos (zlaja(AT)freemail.hu), Apr 27 2005
Partial sums of A002412. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2006
a(n) = (binomial(n+3,n-1)-binomial(n+2,n-2))*(binomial(n+1,n-1)-binomial(n,n-2)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006
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MAPLE
| seq(n^2*(n+1)*(n+2)/6, n=1..50);
A002417:=-(1+3*z)/(z-1)^5; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a[n_] = n*Binomial[n + 2, 3]; Array[a, 37]
From John M. Campbell, May 28, 2011 (Start): Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n + 2, n + 2}], x], x^3], {n, 1, 45}] (End)
Nest[Accumulate, Range[1, 170, 4], 3] (* From Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
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PROG
| (PARI) a(n)=n^2*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Bisection of A002624.
a(n)= A093561(n+3, 4).
Cf. A062196, A002412.
Sequence in context: A195753 A100175 A063489 * A126858 A113751 A107233
Adjacent sequences: A002414 A002415 A002416 * A002418 A002419 A002420
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KEYWORD
| easy,nice,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited and extended by Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 09 2002
Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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