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A075528 Triangular numbers that are half other triangular numbers. 14
0, 3, 105, 3570, 121278, 4119885, 139954815, 4754343828, 161507735340, 5486508657735, 186379786627653, 6331426236682470, 215082112260576330, 7306460390622912753, 248204571168918457275, 8431648959352604634600, 286427860046819639119128 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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.

LINKS

Colin Barker, Table of n, a(n) for n = 0..653

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

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

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

From George F. Johnson, Aug 24 2012: (Start)

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

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

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

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

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

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

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

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

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

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

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

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

(End)

a(n) = 34*a(n-1) -a(n-2) +3, n>=2. - R. J. Mathar, Nov 07 2015

MATHEMATICA

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

PROG

(PARI) concat(0, Vec(3*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ Colin Barker, Jun 18 2015

CROSSREFS

Sequence in context: A103037 A228309 A215945 * A271049 A101485 A226236

Adjacent sequences:  A075525 A075526 A075527 * A075529 A075530 A075531

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower, Sep 19 2002

STATUS

approved

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Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.