%I #28 Feb 22 2023 01:47:03
%S 0,17,51,102,170,255,357,476,612,765,935,1122,1326,1547,1785,2040,
%T 2312,2601,2907,3230,3570,3927,4301,4692,5100,5525,5967,6426,6902,
%U 7395,7905,8432,8976,9537,10115,10710,11322,11951,12597,13260,13940,14637,15351,16082,16830
%N 17 times triangular numbers.
%C Related to the primitive Pythagorean triple [5, 12, 13].
%C Sequence found by reading the line from 0, in the direction 0, 17, ..., and the same line from 0, in the direction 0, 51, ..., in the Pythagorean spiral whose edges have length A195031 and whose vertices are the numbers A195032. This is the main diagonal of the square spiral.
%C Sum of the numbers from 8n to 9n. - _Wesley Ivan Hurt_, Dec 23 2015
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (17*n^2 + 17*n)/2 = 17*n*(n+1)/2 = 17*A000217(n).
%F From _Wesley Ivan Hurt_, Dec 23 2015: (Start)
%F G.f.: 17*x/(1-x)^3.
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
%F a(n) = Sum_{i=8n..9n} i. (End)
%F From _Amiram Eldar_, Feb 22 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 2/17.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/17.
%F Product_{n>=1} (1 - 1/a(n)) = -(17/(2*Pi))*cos(5*Pi/(2*sqrt(17))).
%F Product_{n>=1} (1 + 1/a(n)) = (17/(2*Pi))*cos(3*Pi/(2*sqrt(17))). (End)
%p A195037:=n->17*n*(n+1)/2: seq(A195037(n), n=0..60); # _Wesley Ivan Hurt_, Dec 23 2015
%t 17*Accumulate[Range[0,50]] (* _Harvey P. Dale_, May 31 2014 *)
%t Table[17*n*(n + 1)/2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Dec 23 2015 *)
%o (Magma) [17*n*(n+1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 23 2015
%o (PARI) a(n)=17*n*(n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Bisection of A195032.
%Y Cf. A000217, A195031.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Sep 12 2011
|