A000566 Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2. (Formerly M4358 N1826)

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

Offset

0,3

Comments

Binomial transform of (0, 1, 5, 0, 0, 0, ...). Binomial transform is A084899. - Paul Barry, Jun 10 2003

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

Also the partial sums of A016861, a zero added in front; therefore a(n) = n (mod 5). - R. J. Mathar, Mar 19 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

The n-th heptagonal number equals the sum of the n consecutive integers starting at 2*n-1; for example, 1, 3+4, 5+6+7, 7+8+9+10, etc. In general, the n-th (2k+1)-gonal number is the sum of the n consecutive integers starting at (k-1)*n - (k-2). When k = 1 and 2, this result generates the triangular numbers, A000217, and the pentagonal numbers, A000326, respectively. - Charlie Marion, Mar 02 2022

References

Albert 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.

Leonard 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.

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

S. Barbero, U. Cerruti, and N. Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7., section 4.4.

C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 341

Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae. pp. 137-145.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

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

B. Srinivasa 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. Srinivasa Rao, Heptagonal numbers in the associated Pell sequence and Diophantine equations x^2(5x - 3)^2 = 8y^2 ± 4, Fib. Quarterly, 43 (2005), 302-306.

Leo Tavares, Illustration

Eric Weisstein's World of Mathematics, Heptagonal Number

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: x*(1 + 4*x)/(1 - x)^3. Simon Plouffe in his 1992 dissertation.

a(n) = C(n, 1) + 5*C(n, 2). - Paul Barry, Jun 10 2003

a(n) = Sum_{k = 1..n} (4*n - 3*k). - Paul Barry, Sep 06 2005

a(n) = n + 5*A000217(n-1) - Floor van Lamoen, Oct 14 2005

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for a(0) = 0, a(1) = 1, a(2) = 7. - Jaume Oliver Lafont, Dec 02 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

a(n) = 2*a(n-1) - a(n-2) + 5, with a(0) = 0 and a(1) = 1. - Mohamed Bouhamida, 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(5*n). - 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

Sum_{n>=1} 1/a(n) = sqrt(1 - 2/sqrt(5))*Pi/3 + 5*log(5)/6 - sqrt(5)*log((1 + sqrt(5))/2)/3 = 1.32277925312238885674944226131... . See A244639. - Vaclav Kotesovec, Apr 27 2016

E.g.f.: x*(2 + 5*x)*exp(x)/2. - Ilya Gutkovskiy, Aug 27 2016

From Charlie Marion, May 02 2017: (Start)

a(n+m) = a(n) + 5*n*m + a(m);

a(n-m) = a(n) - 5*n*m + a(m) + 3*m;

a(n) - a(m) = (5*(n + m) - 3)*(n - m)/2.

In general, let P(k,n) be the n-th k-gonal number. Then

P(k,n+m) = P(k,n) + (k - 2)*n*m + P(k,m);

P(k,n-m) = P(k,n) - (k - 2)*n*m + P(k,m) + (k - 4)*m;

P(k,n) - P(k,m) = ((k-2)*(n + m) + 4 - k)*(n - m)/2.

(End)

a(n) = A147875(-n) for all n in Z. - Michael Somos, Jan 25 2019

a(n) = A000217(n-1) + A000217(2*n-1). - Charlie Marion, Dec 19 2019

Product_{n>=2} (1 - 1/a(n)) = 5/7. - Amiram Eldar, Jan 21 2021

a(n) + a(n+1) = (2*n+1)^2 + n^2 - 2*n. In general, if we let P(k,n) = the n-th k-gonal number, then P(k^2-k+1,n)+ P(k^2-k+1,n+1) = ((k-1)*n+1)^2 + (k-2)*(n^2-2*n) = ((k-1)*n+1)^2 + (k-2)*A005563(n-2). When k = 2, this formula reduces to the well-known triangular number formula: T(n) + T(n+1) = (n+1)^2. - Charlie Marion, Jul 01 2021

Example

G.f. = x + 7*x^2 + 18*x^3 + 34*x^4 + 55*x^5 + 81*x^6 + 112*x^7 + ... - Michael Somos, Jan 25 2019

Maple

A000566 := proc(n)
        n*(5*n-3)/2 ;
end proc:
seq(A000566(n), n=0..30); # R. J. Mathar, Oct 02 2020

Mathematica

Table[n (5n - 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* Harvey P. Dale, Oct 13 2011 *)
Join[{0}, Accumulate[Range[1, 315, 5]]] (* Harvey P. Dale, Mar 26 2016 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[7], n], {n, 0, 48}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[7, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)

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
(Python 3) # Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 5, y + 5
A000566 = aList()
print([next(A000566) for i in range(49)]) # Peter Luschny, Aug 04 2019
(Python) [n*(5*n-3)//2 for n in range(50)] # Gennady Eremin, Mar 24 2022

Crossrefs

Cf. A014637, A014640, A014773, A014792, A069099, A131413, A131896, A134483, A000384.

a(n)= A093562(n+1, 2), (5, 1)-Pascal column.

Cf. A006564, A147875, A244639.

Cf. sequences listed in A254963.

Sequence in context: A033537 A352741 A225286 * A225248 A301709 A297646

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