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A000566
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Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.
(Formerly M4358 N1826)
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131
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0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688
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OFFSET
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0,3
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COMMENTS
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Binomial transform of (0,1,5,0,0,0,...) Binomial transform is A084899. - Paul Barry, Jun 10 2003
Also the partial sums of A016861, a zero added in front; therefore a(n) = n (mod 5). - R. J. Mathar, Mar 19 2008
a(n+1) = A153126(n) + n mod 2; a(2*n+1)=A033571(n); a(2*(n+1))=A153127(n)+1. [From Reinhard Zumkeller, Dec 20 2008]
Comment from Ken Rosenbaum, Dec 02 2009: if you multiply the terms of this sequence by 40 and add 9, you get A017354, which is the list of squares of all whole numbers ending in 7 (this is easy to prove).
Also sequence found by reading the line from 0, in the direction 0, 7,..., and the line from 1, in the direction 1, 18,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. These parallel lines are the semi-axes perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011
Also sequence found by reading the line from 0, in the direction 0, 7,... and the parallel line from 1, in the direction 1, 18,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Partial sums give A002413. - Omar E. Pol, Jan 12 2013
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REFERENCES
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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. 2.
B. S. Rao, Heptagonal numbers in the Pell sequence and Diophantine equations 2x^2 = y^2(5y-3)^2 +- 2, Fib. Quarterly, 43 (2005), 194-201.
B. S. Rao, Heptagonal numbers in the associated Pell sequence ..., Fib. Quarterly, 43 (2005), 302-306.
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
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T. D. Noe, Table of n, a(n) for n=0..1000
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 341
_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
Eric Weisstein's World of Mathematics, Heptagonal Number
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FORMULA
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G.f.: x(1+4x)/(1-x)^3; a(n)=C(n, 1)+5C(n, 2). - Paul Barry, Jun 10 2003
a(n)=sum{k=1..n, 4n-3k}. - Paul Barry, Sep 06 2005
a(n)=n+5*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 14 2005
Row sums of triangle A131413 - Gary W. Adamson, Jul 08 2007
Sequence starting (1, 7, 18, 34,...) = binomial transform of (1, 6, 5, 0, 0, 0,...). Also row sums of triangle A131896. - Gary W. Adamson, Jul 24 2007
a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=7 [From Jaume Oliver Lafont, Dec 02 2008]
a(n) = 2*a(n-1) - a(n-2) + 5 with a(0) = 0 and a(1) = 1. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 05 2010]
a(n) = A000217(n)+4*A000217(n-1). [From Vincenzo Librandi, Nov 20 2010]
a(n) = a(n-1)+5*n-4 (with a(0)=0). [From Vincenzo Librandi, Nov 20 2010]
a(n) = A130520(5n). - Philippe Deléham, Mar 26 2013
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MAPLE
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A000566:=-(1+4*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]+5 od: seq(a[n], n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
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s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 0, 6!, 5}]; lst [From Vladimir Joseph Stephan Orlovsky, Oct 25 2008]
Table[n (5n-3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* From Harvey P. Dale, Oct 13 2011 *)
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PROG
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(MAGMA) a000566:=func< n | n*(5*n-3) div 2 >; [ a000566(n): n in [0..50] ];
(PARI) a(n)=n*(5*n-3)/2
(Maxima) makelist(n*(5*n-3)/2, n, 0, 20); /* Martin Ettl, Dec 11 2012 */
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CROSSREFS
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Cf. A014637, A014640, A014773, A014792, A069099, A131413, A131896, A134483, A000384.
a(n)= A093562(n+1, 2), (5, 1)-Pascal column.
n-gonal numbers: A000217, A000290, A000326, A000384, this sequence, A000567, A001106, A001107, A051682, A051624, A051865-A051876.
Sequence in context: A156619 A033537 A225286 * A225248 A169677 A192751
Adjacent sequences: A000563 A000564 A000565 * A000567 A000568 A000569
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KEYWORD
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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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Partially edited by Joerg Arndt, Mar 11 2010
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STATUS
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approved
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