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A069099 Centered heptagonal numbers. 26
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 x + y + 1 <= n. - Reinhard Zumkeller, Jan 23 2012

LINKS

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

E. Weisstein, Centered Polygonal Numbers.

Index entries for sequences related to centered polygonal numbers

Index to sequences with 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)

EXAMPLE

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

For n=2, a(2)=7*2+1-7=8; n=3, a(3)=7*3+8-7=22; n=4, a(4)=7*4+22-7=43. - Vincenzo Librandi, Aug 08 2010

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

CROSSREFS

Cf. A000566 (heptagonal numbers).

Cf. A001263.

Cf. A057655, A001106.

Sequence in context: A058508 A134783 A211529 * A172473 A145067 A112684

Adjacent sequences:  A069096 A069097 A069098 * A069100 A069101 A069102

KEYWORD

easy,nice,nonn

AUTHOR

Terrel Trotter, Jr., Apr 05 2002

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jun 26 2002

STATUS

approved

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Last modified July 28 16:54 EDT 2014. Contains 245003 sequences.