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A000566 Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.
(Formerly M4358 N1826)
144
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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. - 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

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

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).

LINKS

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567

Eric Weisstein's World of Mathematics, Heptagonal Number

FORMULA

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. -  Jaume Oliver Lafont, Dec 02 2008

a(n) = 2*a(n-1) - a(n-2) + 5 with a(0) = 0 and a(1) = 1. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 05 2010

a(n) = A000217(n)+4*A000217(n-1). - Vincenzo Librandi, Nov 20 2010

a(n) = a(n-1)+5*n-4 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010

a(n) = A130520(5n). - Philippe Deléham, Mar 26 2013

a(5*a(n)+11*n+1) = a(5*a(n)+11*n) + a(5*n+1). - Vladimir Shevelev, Jan 24 2014

MAPLE

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, Feb 18 2008

MATHEMATICA

s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 0, 6!, 5}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

Table[n (5n-3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* Harvey P. Dale, Oct 13 2011 *)

PROG

(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 */

(Haskell)

a000566 n = n * (5 * (n - 1) + 2) `div` 2

a000566_list = scanl (+) 0 a016861_list  -- Reinhard Zumkeller, Jun 16 2013

CROSSREFS

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.

Cf. A006564.

Sequence in context: A156619 A033537 A225286 * A225248 A169677 A192751

Adjacent sequences:  A000563 A000564 A000565 * A000567 A000568 A000569

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Partially edited by Joerg Arndt, Mar 11 2010

STATUS

approved

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Last modified September 3 01:21 EDT 2014. Contains 246369 sequences.