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A000326
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Pentagonal numbers: n*(3*n-1)/2.
(Formerly M3818 N1562)
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271
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0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151
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OFFSET
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0,3
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COMMENTS
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The average of the first n (n>0) pentagonal numbers is the n-th triangular number. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
a(n) is the sum of n integers starting from n, i.e. 1, 2+3, 3+4+5, 4+5+6+7, etc. - Jon Perry, Jan 15 2004
Partial sums of 1,4,7,10,13,16,... (1 mod 3), a(2k)=k(6k-1), a(2k-1)=(2k-1)(3k-2) - Jon Perry, Sep 10 2004
a(n) = A126890(n,n-1) for n>0. - Reinhard Zumkeller, Dec 30 2006
Starting with offset 1 = binomial transform of [1, 4, 3, 0, 0, 0,...]. Also, A004736 * [1, 3, 3, 3,...]. - Gary W. Adamson, Oct 25 2007
If Y is a 3-subset of an n-set X then, for n>=4, a(n-3) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
Solutions to the duplication formula 2*a(n)=a(k) are given by the index pairs (n,k) = (5,7), (5577,7887), (6435661,9101399), etc. The indices are integer solutions to the pair of equations 2(6n-1)^2=1+y^2, k=(1+y)/6, so these n can be generated from the subset of numbers [1+A001653(i)]/6, any i, where these are integers, confined to the cases where the associated k=[1+A002315(i)]/6 are also integers. - R. J. Mathar, Feb 01 2008
a(n) is a binomial coefficient C (n, 4) (A000332) if and only if n is a generalized pentagonal number (A001318). Also see A145920. [From Matthew Vandermast, Oct 28 2008]
Let P(n) = pentagonal number, T(n) = triangular number, then P(n)= T(n)+2*T(n-1) [From Vincenzo Librandi, Nov 20 2010]
Even octagonal numbers divided by 8. - Omar E. Pol, Aug 18 2011
Sequence found by reading the line from 0, in the direction 0, 5,... and the line from 1, in the direction 1, 12,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
The hyper-Wiener index of the star-tree with n edges (see A196060, Example). - Emeric Deutsch, Sep 30 2011
More generally the n-th k-gonal number is equal to n + (k-2)*A000217(n-1), n >= 1, k >= 3. In this case k = 5. - Omar E. Pol, Apr 06 2013
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REFERENCES
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G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
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. 1.
R. T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 83-87.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.
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).
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 186.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 98-100 Penguin Books 1987.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
J. Bell, Euler and the pentagonal number theorem
L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 1
L. Euler, Observatio de summis divisorum p. 8.
L. Euler, An observation on the sums of divisors p. 8.
L. Euler, On the remarkable properties of the pentagonal numbers
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 339
Hyun Kwang Kim, On Regular Polytope Numbers
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326
Eric Weisstein's World of Mathematics, Pentagonal Number
Wikipedia, Pentagonal number
Index entries for "core" sequences
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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Product_{m>0} (1-q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry, Jul 20 2003
G.f.: x*(1+2*x)/(1-x)^3.
E.g.f.: exp(x)*(x+3*x^2/2).
a(n) = n*(3*n-1)/2.
a(-n) = A005449(n).
a(n) = binomial(3*n, 2)/3. - Paul Barry, Jul 20 2003
a(n) = A000290(n) + A000217(n-1). - Lekraj Beedassy, Jun 07 2004
a(0) = 0, a(1) = 1; for n >= 2, a(n) = 2*a(n-1)- a(n-2) + 3. - Miklos Kristof, Mar 09 2005
a(n) = sum{k=1..n, 2n-k}; - Paul Barry, Aug 19 2005
a(n) = 3*A000217(n) - 2*n . - Lekraj Beedassy, Sep 26 2006
a(n) = A049452(n)-A022266(n), a(n) = A033991(n)-A005476(n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007
Equals A034856(n) + (n - 1)^2. Also equals A051340 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
a(n) = C(n+1,2) + 2 C(n,2).
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=5. [Jaume Oliver Lafont, Dec 02 2008]
a(n) = a(n-1)+3*n-2 with n>0, a(0)=0. [Vincenzo Librandi, Nov 20 2010]
a(n) = A014642(n)/8. - Omar E. Pol, Aug 18 2011
a(n) = A142150(n) + A191967(n). - Reinhard Zumkeller, Jul 07 2012
a(n) = (A000290(n) + A000384(n))/2 = (A000217(n) + A000566(n))/2 = A049450(n)/2. - Omar E. Pol, Jan 11 2013
a(n) = n*A000217(n)-(n-1)*A000217(n-1). [Bruno Berselli, Jan 18 2013]
a(n) = A005449(n) - n. - Philippe Deléham, Mar 30 2013
From Oskar Wieland, Apr 10 2013: (Start)
a(n) = a(n+1)-A016777(n),
a(n) = a(n+2)-A016969(n),
a(n) = a(n+3)-A016777(n)*3 = a(n+3)-A017197(n),
a(n) = a(n+4)-A016969(n)*2 = a(n+4)-A017641(n),
a(n) = a(n+5)-A016777(n)*5,
a(n) = a(n+6)-A016969(n)*3,
a(n) = a(n+7)-A016777(n)*7,
a(n) = a(n+8)-A016969(n)*4,
a(n) = a(n+9)-A016777(n)*9. (End)
a(n) = A000217(2n-1) - A000217(n-1), for n>0. - Ivan N. Ianakiev, Apr 17 2013
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EXAMPLE
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Illustration of initial terms:
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. o o
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. o o o o o o o o o o o o o o o
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. 1 5 12 22 35 - Philippe Deléham, Mar 30 2013
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MAPLE
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A000326 := n->n*(3*n-1)/2;
A000326:=-(1+2*z)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+3 od: seq(a[n], n=0..50); # Miklos Kristof - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
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Table[n(3n - 1)/2, {n, 0, 60}] - Stefan Steinerberger, Apr 01 2006
s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 1, 5!, 3}]; lst (* _Vladimir Orlovsky, Apr 02 2009 *)
Array[ #*(3*# - 1)/2 &, 47, 0] (* Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009 *)
LinearRecurrence[{3, -3, 1}, {0, 1, 5}, 61] (* Harvey P. Dale, Dec 27 2011 *)
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PROG
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(PARI) a(n)=n*(3*n-1)/2
(Haskell)
a000326 n = n * (3 * n - 1) `div` 2 -- Reinhard Zumkeller, Jul 07 2012
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CROSSREFS
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The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
Cf. A001318 (generalized pentagonal numbers), A005449, A049050, A033570, A010815, A034856, A051340, A004736.
Cf. n-gonal numbers: A000217, A000290, this sequence, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.
Cf. A033568, A049453.
Cf. A002411 (partial sums), A033579.
Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
Sequence in context: A131976 A074376 A134340 * A022795 A025734 A153818
Adjacent sequences: A000323 A000324 A000325 * A000327 A000328 A000329
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KEYWORD
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core,nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Removed incorrect example Joerg Arndt, Mar 11 2010
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STATUS
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approved
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