A051874 22-gonal numbers: a(n) = n*(10*n-9).

0, 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332, 1573, 1834, 2115, 2416, 2737, 3078, 3439, 3820, 4221, 4642, 5083, 5544, 6025, 6526, 7047, 7588, 8149, 8730, 9331, 9952, 10593, 11254, 11935, 12636, 13357, 14098, 14859

Offset: 0

Author

N. J. A. Sloane, Dec 15 1999

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 22,... and the parallel line from 1, in the direction 1, 63,..., in the square spiral whose vertices are the generalized 22-gonal numbers. - Omar E. Pol, Jul 18 2012

Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 22,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 29 2012

This is also a star hendecagonal number: a(n) = A051682(n) + 11*A000217(n-1). - Luciano Ancora, Mar 30 2015

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+20 od: seq(a[n], n=0..39); # Zerinvary Lajos, Feb 18 2008

Mathematica

Table[n (10 n -9), {n, 0, 40}] (* Harvey P. Dale, Sep 19 2011 *)
CoefficientList[Series[x (1 + 19 x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)

PARI

a(n)=n*(10*n-9) \\ Charles R Greathouse IV, Jan 24 2014

Formula

a(n) = 2*a(n-1)-a(n-2)+20 with n>1, a(0)=0, a(1)=1. - Zerinvary Lajos, Feb 18 2008

a(n) = 20*n+a(n-1)-19 with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010

G.f.: x*(1+19*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(20*a(n)+191*n+1) = a(20*a(n)+191*n) + a(20*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 10/11. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 10*x^2). - Nikolaos Pantelidis, Feb 05 2023