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%I A011916 #47 Jul 26 2018 18:41:16
%S A011916 0,3,44,615,8568,119339,1662180,23151183,322454384,4491210195,
%T A011916 62554488348,871271626679,12135248285160,169022204365563,
%U A011916 2354175612832724,32789436375292575,456697933641263328,6360981634602394019
%N A011916 a(n) = ((b(n)-1)+sqrt(3*b(n)^2-4*b(n)+1))/2, where b(n) is A011922(n).
%C A011916 Integers k such that k^2 = Sum_{i=1..x} (k+i) for some value of x. 3 is a term because 3^2=9 and 4+5=9; 44 is a term because 44^2=1936 and the sum of (45,46,47,...,76) = 1936. - _Gil Broussard_, Dec 23 2008
%C A011916 Also the index of the first of two consecutive octagonal numbers whose sum is equal to the sum of two consecutive squares. - _Colin Barker_, Dec 20 2014
%C A011916 Also the index of a triangular number included in A239071. - _Ivan Neretin_, May 31 2015
%D A011916 Mario Velucchi, "Seeing couples" in Recreational and Educational Computing, to appear 1997. [apparently never materialized, _Colin Barker_, Dec 23 2014]
%H A011916 Colin Barker, <a href="/A011916/b011916.txt">Table of n, a(n) for n = 0..874</a>
%H A011916 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F A011916 From _R. J. Mathar_, Apr 15 2010: (Start)
%F A011916 a(n) = +15*a(n-1) -15*a(n-2) +a(n-3).
%F A011916 G.f.: x*(-3 + x) / ((x - 1)*(x^2 - 14*x + 1)). (End)
%F A011916 From _Michael Somos_, Jul 27 2012: (Start)
%F A011916 a(n) = A109437(2*n).
%F A011916 a(-1 - n) = -A109437(2*n + 1). (End)
%F A011916 a(n) = (A001353(n+1)^2 - A001075(n)^2)/4. - _Richard R. Forberg_, Aug 26 2013
%F A011916 a(n) = (-2-(7-4*sqrt(3))^n*(-1+sqrt(3))+(1+sqrt(3))*(7+4*sqrt(3))^n)/12. - _Colin Barker_, Mar 05 2016
%t A011916 RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 0, a[1] == 3, a[2] == 44}, a, {n, 0, 17}] (* _Michael De Vlieger_, Jul 02 2015 *)
%t A011916 LinearRecurrence[{15,-15,1},{0,3,44},30] (* _Harvey P. Dale_, Jul 26 2018 *)
%o A011916 (PARI) {a(n) = if( n<0, n = -n; polcoeff( x*(1 - 3*x) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n), polcoeff( x*(x - 3) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n))} /* _Michael Somos_, Jul 27 2012 */
%o A011916 (PARI) concat(0, Vec(x*(-3+x)/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ _Colin Barker_, Dec 20 2014
%Y A011916 Cf. A011918, A109437.
%K A011916 nonn,easy
%O A011916 0,2
%A A011916 Mario Velucchi (mathchess(AT)velucchi.it)
%E A011916 More terms from _R. J. Mathar_, Apr 15 2010
%E A011916 Added a(0)=0, _Michael Somos_, Jul 27 2012

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