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A006000
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a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4.
(Formerly M3436)
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4
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1, 4, 12, 28, 55, 96, 154, 232, 333, 460, 616, 804, 1027, 1288, 1590, 1936, 2329, 2772, 3268, 3820, 4431, 5104, 5842, 6648, 7525, 8476, 9504, 10612, 11803, 13080, 14446, 15904, 17457, 19108, 20860, 22716, 24679, 26752, 28938, 31240, 33661, 36204, 38872, 41668, 44595, 47656, 50854, 54192
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Enumerates certain paraffins.
a(n) is the (n+1)st (n+3)-gonal number. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 20 2001
Sum of n terms of an arithmetic progression with the first term 1 and the common difference n: a(1)=1 a(2) = 1+3 a(3) = 1+4+7 a(4) = 1+5+9+13 etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2004
This is identical to: first triangular number A000217, 2nd square number A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 5th heptagonal number A000566, 6th octagonal number A000567, ..., (n+1)-th (n+3)-gonal number = main diagonal of rectangular array T(n,k) of polygonal numbers, by diagonals, referred to in A086271. - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 19 2007
Also (n + 1)! times the determinant of the n X n matrix given by m(i,j) = (i+1)/i
if i=j and otherwise 1. For example, (6 + 1)!*Det[{{2,1,1,1,1,1}, {1,3/2,1,1,1,1},{1,1,4/3,1,1,1}, {1,1,1,5/4,1,1}, {1,1,1,1,6/5,1}, {1,1,1,1,1,7/6}}] =154 =a(6) - John M. Campbell, May 20, 2011
a(n-1)=:N_2(n), n>=1, is the number of 2-faces of n planes in generic position in three-dimensional space. See comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p.506. [Wolfdieter Lang, May 27 2011]
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REFERENCES
| V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_2.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [See p. 301]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| William A. Tedeschi, Table of n, a(n) for n=1..10000
Index entries for sequences related to linear recurrences with constant coefficients
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.
Eric Weisstein's World of Mathematics, Polygonal Number
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FORMULA
| a(n) = sum(j=1..n+1, binomial(0,0*j)+binomial(n+1,2) ). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 25 2006
a(n-1) = n + (n^3 - n^2)/2 = n + n*T(n-1) where T(n-1) is a triangular number, n>=1. [From William A. Tedeschi (fynmun(AT)att.net), Aug 22 2010]
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MAPLE
| A006000:=(1+2*z**2)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a[n_]:=(n^3-n^2)/2+n; Table[a[n], {n, 5!}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 07 2010]
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PROG
| (Python) a = lambda n: a = lambda n: n + ((n**3 - n**2)//2) [From William A. Tedeschi (fynmun(AT)att.net), Aug 22 2010]
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CROSSREFS
| Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A086271.
Sequence in context: A112087 A166019 A184633 * A161216 A085622 A011940
Adjacent sequences: A005997 A005998 A005999 * A006001 A006002 A006003
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001
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