A051870 - OEIS

%I #81 May 04 2021 01:04:17

%S 0,1,18,51,100,165,246,343,456,585,730,891,1068,1261,1470,1695,1936,

%T 2193,2466,2755,3060,3381,3718,4071,4440,4825,5226,5643,6076,6525,

%U 6990,7471,7968,8481,9010,9555,10116,10693,11286,11895,12520

%N 18-gonal (or octadecagonal) numbers: a(n) = n*(8*n-7).

%C Also, sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Apr 26 2008

%C This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - _T. D. Noe_, Apr 13 2011

%C Also sequence found by reading the line from 0, in the direction 0, 18, ... and the parallel line from 1, in the direction 1, 51, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - _Omar E. Pol_, Jul 18 2012

%C Partial sums of 16n + 1 (with offset 0), compare A005570. - _Jeremy Gardiner_, Aug 04 2012

%C All x values for Diophantine equation 32*x + 49 = y^2 are given by this sequence and A139278. - _Bruno Berselli_, Nov 11 2014

%C This is also a star enneagonal number: a(n) = A001106(n) + 9*A000217(n-1). - _Luciano Ancora_, Mar 30 2015

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

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

%H Jeremy Gardiner, <a href="/A051870/b051870.txt">Table of n, a(n) for n = 0..999</a>

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal number</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 G.f.: x*(1+15*x)/(1-x)^3. - _Bruno Berselli_, Feb 04 2011

%F a(n) = 16*n + a(n-1) - 15, with n > 0, a(0) = 0. - _Vincenzo Librandi_, Aug 06 2010

%F a(16*a(n)+121*n+1) = a(16*a(n)+121*n) + a(16*n+1). - _Vladimir Shevelev_, Jan 24 2014

%F E.g.f.: (8*x^2 + x)*exp(x). - _G. C. Greubel_, Jul 18 2017

%F Sum_{n>=1} 1/a(n) = ((1+sqrt(2))*Pi + 2*sqrt(2)*arccoth(sqrt(2)) + 8*log(2))/14. - _Amiram Eldar_, Oct 20 2020

%F Product_{n>=2} (1 - 1/a(n)) = 8/9. - _Amiram Eldar_, Jan 22 2021

%p A051870 := proc(n) n*(8*n-7) ; end proc: seq(A051870(n),n=0..30) ; # _R. J. Mathar_, Feb 05 2011

%t Table[n (8 n - 7), {n, 0, 40}] (* _Bruno Berselli_, Nov 11 2014 *)

%o (PARI) a(n)=n*(8*n-7) \\ _Charles R Greathouse IV_, Jul 19 2011

%Y Cf. A014634, A014635, A033585, A033586, A033587, A035008, A069129, A085250, A129271-A129278, A139278.

%K nonn,easy

%O 0,3

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