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A002411
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Pentagonal pyramidal numbers: n^2*(n+1)/2.
(Formerly M4116 N1709)
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63
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0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, 11368, 12615, 13950, 15376, 16896, 18513, 20230, 22050, 23976, 26011, 28158, 30420, 32800, 35301, 37926, 40678
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors. - R. H. Hardin (rhhardin(AT)att.net), Feb 23 2002
a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
Also as a(n)=(1/6)*(3*n^3+3*n^2), n>0: structured trigonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 18 2005
If Y is a 3-subset of an n-set X then, for n>=5, a(n-4) is the number of 5-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007
a(n-1), n>=2, is the number of ways to have n identical objects in m=2 of altogether n distinguishable boxes (n-2 boxes stay empty). W. Lang, Nov 13 2007.
a(n+1) is the convolution of (n+1) and (3n+1). [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2008]
a(n+1) = denom(2/((n+2)*(n+1)^2)) [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009]
The number of 3-character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.
Partial sums are A001296 4-dimensional pyramidal numbers: (3n+1)*C(n+2,3)/4 [Jonathan Vos Post, March 26, 2011]
a(n-1):=N_1(n), n>=1, is the number of edges of n planes in generic position in three-dimensional space. See a 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_1.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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=0..1000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
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, Pentagonal Pyramidal Number.
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| Average of n^2 and n^3.
a(n) = sum of n smallest multiples of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 20 2002
G.f.: x*(1+2*x)/(1-x)^4 - Paul Barry (pbarry(AT)wit.ie), Mar 21 2003
a(n) = sum(k=0..n, n*(n-k) ). - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
a(n) = whole numbers * triangular numbers - Xavier Acloque Oct 27 2003
a(n) = (1/2)*Sum[Sum[(i+j),{i, 1, n}],{j, 1, n}] = (1/2)*(n^2+n^3) = (1/2)*A011379(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006
a(n) = sum(j=0..n, n*j). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 12 2006
Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0,...] = (1, 6, 18, 40, 75,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2007
G.f.: x*F(2,3;1;x); [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2008]
sum(j>=1, 1/a(j)) = hypergeom([1, 1, 1], [2, 3], 1) = -2+2*Zeta(2) [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009]
a(0)=0, a(1)=1, a(2)=6, a(3)=18, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) [From Harvey P. Dale, Oct 20 2011]
a(n) = n*binomial(n+1,2) = n*A000217(n). [Arkadiusz Wesolowski, Feb 10 2012]
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EXAMPLE
| a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!)+ 2!/(2!) ) = 6*(2+1) =18 ways. The m=2 part partitions of 4, namely (1,3) and (2,2), specify the filling of each of the 6 possible two box choices. W. Lang, Nov 13 2007.
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MAPLE
| seq(n^2*(n+1)/2, n=0..40);
A002411:=(1+2*z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[n^2*(n+1)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 18}, 50] (* From Harvey P. Dale, Oct 20 2011 *)
Nest[Accumulate, Range[1, 140, 3], 2]] (* From Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
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PROG
| (PARI) a(n)=n^2*(n+1)/2
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CROSSREFS
| Cf. A001296, A015223, A015224, A014799, A014800.
Cf. A011379, A127739, A132118.
A006002(n)=-a(-1-n).
a(n)= A093560(n+2, 3), (3, 1)-Pascal column.
A row or column of A132191.
Second column of triangle A103371.
Sequence in context: A129863 A035489 A122061 * A023658 A059834 A015224
Adjacent sequences: A002408 A002409 A002410 * A002412 A002413 A002414
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| zeta hypergeometric formula [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009]
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