A139277 - OEIS

%I #39 Dec 24 2024 22:12:35

%S 0,13,42,87,148,225,318,427,552,693,850,1023,1212,1417,1638,1875,2128,

%T 2397,2682,2983,3300,3633,3982,4347,4728,5125,5538,5967,6412,6873,

%U 7350,7843,8352,8877,9418,9975,10548,11137,11742,12363,13000

%N a(n) = n*(8*n+5).

%C Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139273 in the same spiral.

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

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

%F a(n) = 8*n^2 + 5*n.

%F Sequences of the form a(n) = 8*n^2 + c*n have generating functions x*{c+8 + (8-c)*x}/(1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - _R. J. Mathar_, May 12 2008

%F a(n) = 16*n + a(n-1) - 3 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F Sum_{n>=1} 1/a(n) = (sqrt(2)-1)*Pi/10 - 4*log(2)/5 + sqrt(2)*log(sqrt(2)+1)/5 + 8/25. - _Amiram Eldar_, Mar 18 2022

%F E.g.f.: exp(x)*x*(13 + 8*x). - _Elmo R. Oliveira_, Dec 15 2024

%t Table[n (8 n + 5), {n, 0, 50}] (* _Bruno Berselli_, Aug 22 2018 *)

%t LinearRecurrence[{3,-3,1},{0,13,42},50] (* _Harvey P. Dale_, Dec 04 2018 *)

%o (PARI) a(n)=n*(8*n+5) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250.

%Y Cf. A139271, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Apr 26 2008