

A011916


a(n) = ((b(n)1)+sqrt(3*b(n)^24*b(n)+1))/2, where b(n) is A011922(n).


11



0, 3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384, 4491210195, 62554488348, 871271626679, 12135248285160, 169022204365563, 2354175612832724, 32789436375292575, 456697933641263328, 6360981634602394019
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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(n1) 15*a(n2) +a(n3).
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(74*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) / ((x1) * (x^2  14*x + 1)) + x * O(x^n), n), polcoeff( x*(x  3) / ((x1) * (x^2  14*x + 1)) + x * O(x^n), n))} /* Michael Somos, Jul 27 2012 */
(PARI) concat(0, Vec(x*(3+x)/((x1)*(x^214*x+1)) + O(x^100))) \\ Colin Barker, Dec 20 2014


CROSSREFS

Cf. A011918, A109437.
Sequence in context: A046946 A327360 A092545 * A337249 A259785 A193623
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



