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A075528 Triangular numbers that are half other triangular numbers. 15
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

LINKS

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

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

H. J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)

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.

8*a(n)+1 = A000129(2*n+1)^2.

16*a(n)+1 = A002315(n)^2.

128*a(n)^2+24*a(n)+1 is a perfect square.

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

a(n) = A000217(A053141(n)). - R. J. Mathar, Aug 16 2019

a(n) = (a(n-1)*(a(n-1)-3))/a(n-2) for n>2. - Vladimir Pletser, Apr 08 2020

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

Cf. A000129, A000217, A001652, A002315, A029546, A029549, A046090, A046729, A053141, A084159, A096979.

Sequence in context: A103037 A228309 A215945 * A334776 A271049 A101485

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 September 23 19:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)