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A075528 Triangular numbers that are half other triangular numbers. 14

%I

%S 0,3,105,3570,121278,4119885,139954815,4754343828,161507735340,

%T 5486508657735,186379786627653,6331426236682470,215082112260576330,

%U 7306460390622912753,248204571168918457275,8431648959352604634600,286427860046819639119128

%N Triangular numbers that are half other triangular numbers.

%C This is the sequence of 1/2 the areas, x(n)*y(n)/2, of the ordered Pythagorean triples (x(n),y(n)=x(n)+1,z(n)) with x(0)=0,y(0)=1,z(0)=1,a(0)=0 and x(1)=3,y(1)=4,z(1)=5,a(1)=3. - _George F. Johnson_, Aug 24 2012

%D Martin V Bonsangue, Gerald E Gannon & Laura J. Pheifer, Misinterpretations can sometimes be a good thing, Math. Teacher, vol. 95, No. 6 (2002) pp. 446-449.

%H Colin Barker, <a href="/A075528/b075528.txt">Table of n, a(n) for n = 0..653</a>

%H H. J. Hindin, <a href="/A006062/a006062.pdf">Stars, hexes, triangular numbers and Pythagorean triples</a>, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).

%F a(n) = 3*A029546(n-1) = A029549(n)/2.

%F G.f.: 3*x/((1-x)*(1-34*x+x^2)).

%F From _George F. Johnson_, Aug 24 2012: (Start)

%F a(n) = ((3+2*sqrt(2))^(2*n+1)+(3-2*sqrt(2))^(2*n+1)-6)/64.

%F 16*a(n)+1 = (A002315(n))^2 , 8*a(n)+1=(A000129(2*n+1))^2,

%F 128*a(n)^2+24*a(n)+1 are perfect squares.

%F a(n+1) = 17*a(n)+3/2+3*sqrt((8*a(n)+1)*(16*a(n)+1))/2.

%F a(n-1) = 17*a(n)+3/2-3*sqrt((8*a(n)+1)*(16*a(n)+1))/2.

%F a(n-1)*a(n+1) = a(n)*(a(n)-3); a(n+1)=34*a(n)-a(n-1)+3.

%F a(n+1) = 35*a(n)-35*a(n-1)+a(n-2); a(n)=A096979(2*n)/2.

%F a(n) = A084159(n)*A046729(n)/4 = A001652(n)*A046090(n)/4.

%F Lim. as n -> inf. of a(n)/a(n-1) = 17+12*sqrt(2).

%F Lim. as n -> inf. of a(n)/a(n-2) = (17+12*sqrt(2))^2 = 577+408*sqrt(2).

%F Lim. as n -> inf. of a(n)/a(n-r) = (17+12*sqrt(2))^r.

%F Lim. as n -> inf. of a(n-r)/a(n) = (17+12*sqrt(2))^(-r)=(17-12*sqrt(2))^r.

%F (End)

%F a(n) = 34*a(n-1) -a(n-2) +3, n>=2. - _R. J. Mathar_, Nov 07 2015

%t CoefficientList[ Series[ 3x/(1 - 35 x + 35 x^2 - x^3), {x, 0, 15}], x] (* _Robert G. Wilson v_, Jun 24 2011 *)

%o (PARI) concat(0, Vec(3*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ _Colin Barker_, Jun 18 2015

%K nonn,easy

%O 0,2

%A _Christian G. Bower_, Sep 19 2002

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Last modified December 11 19:37 EST 2018. Contains 318049 sequences. (Running on oeis4.)