%I #53 Feb 06 2023 07:06:07

%S 0,1,19,54,106,175,261,364,484,621,775,946,1134,1339,1561,1800,2056,

%T 2329,2619,2926,3250,3591,3949,4324,4716,5125,5551,5994,6454,6931,

%U 7425,7936,8464,9009,9571,10150,10746,11359,11989,12636,13300

%N 19-gonal (or enneadecagonal) numbers: n(17n-15)/2.

%C Sequence found by reading the line from 0, in the direction 0, 19, ... and the parallel line from 1, in the direction 1, 54, ..., in the square spiral whose vertices are the generalized 19-gonal numbers. - _Omar E. Pol_, Jul 18 2012

%C Partial sums of A215137 (17n + 1). - _Jeremy Gardiner_, Aug 04 2012

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.

%D Elena Deza and Michel M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

%H Jeremy Gardiner, <a href="/A051871/b051871.txt">Table of n, a(n) for n = 0..999</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = n(17n-15)/2.

%F G.f.: x*(1+16*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011

%F a(n) = 17*n + a(n-1) - 16 (with a(0) = 0). - _Vincenzo Librandi_, Aug 06 2010

%F a(17*a(n) + 137*n + 1) = a(17*a(n) + 137*n) + a(17*n+1). - _Vladimir Shevelev_, Jan 24 2014

%F Product_{n>=2} (1 - 1/a(n)) = 17/19. - _Amiram Eldar_, Jan 22 2021

%F E.g.f.: exp(x)*(x + 17*x^2/2). - _Nikolaos Pantelidis_, Feb 06 2023

%e a(1) = 17 * 1 + 0 - 16 = 1.

%e a(2) = 17 * 2 + 1 - 16 = 19.

%e a(3) = 17 * 3 + 19 - 16 = 54. - _Vincenzo Librandi_, Aug 06 2010

%p A051871 := proc(n) n*(17*n-15)/2 ; end proc: seq(A051871(n),n=0..30) ; # _R. J. Mathar_, Feb 05 2011

%t Table[(17n^2 - 15n)/2, {n, 0, 39}] (* _Alonso del Arte_, Feb 19 2015 *)

%o (PARI) a(n)=n*(17*n-15)/2; \\ _Charles R Greathouse IV_, Jan 24 2014

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 15 1999