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Search: id:A006000
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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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+0
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; listen)
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OFFSET
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0,2
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COMMENT
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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: 1st 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 (jvospost2(AT)yahoo.com), Dec 19 2007
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REFERENCES
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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]
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LINKS
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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.
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FORMULA
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a:=n->sum((binomial(0,0*j)+binomial(n+1,2)),j=1..n+1): seq(a(n), n>=1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 25 2006
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MAPLE
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a:=n->sum((binomial(0, 0*j)+binomial(n+1, 2)), j=1..n+1): seq(a(n), n=1..49); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 25 2006
seq(add(k+add(l, k=0..n), l=0..n), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 04 2007
A006000:=(1+2*z**2)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
with (combinat):seq((fibonacci(4, n)-n^2)/2, n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008
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CROSSREFS
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Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A086271.
Sequence in context: A009906 A109629 A112087 this_sequence A085622 A011940 A102653
Adjacent sequences: A005997 A005998 A005999 this_sequence A006001 A006002 A006003
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KEYWORD
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nonn,easy
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AUTHOR
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njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001
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