A257709 - OEIS

%I #15 Sep 08 2022 08:46:12

%S 8,14,39,53,103,112,206,264,509,647,1141,1230,2160,2734,5159,6525,

%T 11415,12296,21502,27184,51189,64711,113117,121838,212968,269214,

%U 506839,640693,1119863,1206192,2108286,2665064,5017309,6342327,11085621,11940190,20870000

%N Numbers n such that T(n) + T(n+1) + ... + T(n+26) is a square, where T = A000217 (triangular numbers).

%C Positive integers y in the solutions to 2*x^2-27*y^2-729*y-6552 = 0.

%H Colin Barker, <a href="/A257709/b257709.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,10,-10,0,0,0,0,-1,1).

%F G.f.: x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)).

%t LinearRecurrence[{1, 0, 0, 0, 0, 10, -10, 0, 0, 0, 0, -1, 1}, {8, 14, 39, 53, 103, 112, 206, 264, 509, 647, 1141, 1230, 2160}, 50] (* _Vincenzo Librandi_, May 05 2015 *)

%t Position[Total/@Partition[Accumulate[Range[70000]],27,1],_?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* The program generates the first 22 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* _Harvey P. Dale_, Jul 27 2021 *)

%o (PARI) Vec(x*(2*x^12+x^11+6*x^10+2*x^9+5*x^8+2*x^7-14*x^6-9*x^5-50*x^4-14*x^3-25*x^2-6*x-8) / ((x-1)*(x^12-10*x^6+1)) + O(x^100))

%o (Magma) I:=[8,14,39,53,103,112,206,264,509,647,1141,1230, 2160]; [n le 13 select I[n] else Self(n-1)+10*Self(n-6)-10*Self(n-7)-Self(n-12)+Self(n-13): n in [1..40]]; // _Vincenzo Librandi_, May 05 2015

%Y Cf. A176541, A176542, A000217, A000292, A001110, A077415.

%Y Cf. A116476 (length 11), A257293 (length 13), A257707 (length 23), A257708 (length 25), A257710 (length 37).

%K nonn,easy

%O 1,1

%A _Colin Barker_, May 04 2015