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A069099 Centered heptagonal numbers. 54
1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals the triangular numbers convolved with [ 1, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009

Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and abs(x + y) < n, counting twice pairs of equal numbers. - Reinhard Zumkeller, Jan 23 2012

The number of pairs without repetitions is a(n) - 2n + 3 for n > 1. For example, there are 19 such pairs for n = 3: (-2, 0), (-2, 1), (-2, 2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (1, -2), (1, -1), (1, 0), (1, 1), (2, -2), (2, -1), (2, 0).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.

Index entries for sequences related to centered polygonal numbers

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

FORMULA

a(n) = (7*n^2 - 7*n + 2)/2.

a(n) = 1 + Sum_{k=1..n} 7*k. - Xavier Acloque, Oct 26 2003

Binomial transform of [1, 7, 7, 0, 0, 0,...]; Narayana transform (A001263) of [1, 7, 0, 0, 0,...]. - Gary W. Adamson, Dec 29 2007

a(n) = 7*n+a(n-1)-7 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

G.f.: x*(1+5*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011

a(0)=1, a(1)=8, a(2)=22, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Jun 04 2011

a(n) = A024966(n-1) + 1. - Omar E. Pol, Oct 03 2011

a(n) = 2*a(n-1) - a(n-2) + 7. - Ant King, Jun 17 2012

From Ant King, Jun 17 2012: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi/sqrt(7)*tanh(Pi/(2*sqrt(7))) = 1.264723171685652...

a(n) == 1 (mod 7) for all n.

The sequence of digital roots of the a(n) is period 9: repeat[1, 8, 4, 7, 8, 7, 4, 8, 1] (the period is a palindrome).

The sequence of a(n) mod 10 is period 20: repeat [1, 8, 2, 3, 1, 6, 8, 7, 3, 6, 6, 3, 7, 8, 6, 1, 3, 2, 8, 1] (the period is a palindrome).

(End)

E.g.f.: -1 +  (2 + 7*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jun 30 2016

a(n) = A101321(7,n-1). - R. J. Mathar, Jul 28 2016

EXAMPLE

a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.

From Bruno Berselli, Oct 27 2017: (Start)

1   =         -(0) + (1).

8   =       -(0+1) + (2+3+4).

22  =     -(0+1+2) + (3+4+5+6+7).

43  =   -(0+1+2+3) + (4+5+6+7+8+9+10).

71  = -(0+1+2+3+4) + (5+6+7+8+9+10+11+12+13). (End)

MATHEMATICA

FoldList[#1 + #2 &, 1, 7 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)

LinearRecurrence[{3, -3, 1}, {1, 8, 22}, 50] (* Harvey P. Dale, Jun 04 2011 *)

PROG

(Haskell)

a069099 n = length

   [(x, y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n - 1]

-- Reinhard Zumkeller, Jan 23 2012

(PARI) a(n)=(7*n^2-7*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A000566 (heptagonal numbers).

Cf. A001263, A057655, A001106.

Sequence in context: A058508 A134783 A211529 * A172473 A145067 A112684

Adjacent sequences:  A069096 A069097 A069098 * A069100 A069101 A069102

KEYWORD

nonn,easy,nice

AUTHOR

Terrel Trotter, Jr., Apr 05 2002

EXTENSIONS

Comment corrected and extended by Mauro Fiorentini, Jan 01 2018

STATUS

approved

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Last modified July 19 13:05 EDT 2018. Contains 312775 sequences. (Running on oeis4.)