A011916 a(n) = ((b(n)-1)+sqrt(3*b(n)^2-4*b(n)+1))/2, where b(n) is A011922(n).

0, 3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384, 4491210195, 62554488348, 871271626679, 12135248285160, 169022204365563, 2354175612832724, 32789436375292575, 456697933641263328, 6360981634602394019

Offset

0,2

Comments

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

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

Also the index of a triangular number included in A239071. - Ivan Neretin, May 31 2015

References

Mario Velucchi, "Seeing couples" in Recreational and Educational Computing, to appear 1997. [apparently never materialized, Colin Barker, Dec 23 2014]

Links

Colin Barker, Table of n, a(n) for n = 0..874

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Formula

From R. J. Mathar, Apr 15 2010: (Start)

a(n) = +15*a(n-1) -15*a(n-2) +a(n-3).

G.f.: x*(-3 + x) / ((x - 1)*(x^2 - 14*x + 1)). (End)

From Michael Somos, Jul 27 2012: (Start)

a(n) = A109437(2*n).

a(-1 - n) = -A109437(2*n + 1). (End)

a(n) = (A001353(n+1)^2 - A001075(n)^2)/4. - Richard R. Forberg, Aug 26 2013

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

Mathematica

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 *)
LinearRecurrence[{15, -15, 1}, {0, 3, 44}, 30] (* Harvey P. Dale, Jul 26 2018 *)

Prog

(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 */
(PARI) concat(0, Vec(x*(-3+x)/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, Dec 20 2014

Crossrefs

Cf. A011918, A109437.

Sequence in context: A092545 A353878 A359631 * A337249 A259785 A369944

Adjacent sequences: A011913 A011914 A011915 * A011917 A011918 A011919

Keyword

nonn,easy

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Extensions

More terms from R. J. Mathar, Apr 15 2010

Added a(0)=0, Michael Somos, Jul 27 2012

Status

approved