login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A200993 Triangular numbers, T(m), that are two-thirds of another triangular number; T(m) such that 3*T(m) = 2*T(k) for some k. 7
0, 10, 990, 97020, 9506980, 931587030, 91286021970, 8945098566040, 876528373449960, 85890835499530050, 8416425350580494950, 824723793521388975060, 80814515339745539060940, 7918997779501541438997070, 775980967875811315482651930, 76038215854050007375860892080 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n>1, a(n) = 98*a(n-1) - a (n-2) + 10.  In general, for m>0, let b(n) be those triangular numbers such that for some triangular number c(n), (m+1)*b(n) = m*c(n).  Then b(0) = 0, b(1)= A014105(m) and for n>1,  b(n) = 2*A069129(m+1)*b(n-1) - b(n-2) + A014105(m).

Further, c(0) = 0, c(1) = A000384(m+1) and for n>1, c(n) = 2*A069129(m+1)*c(n-1) - c(n-2) + A000384(m+1).

LINKS

Colin Barker, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

FORMULA

G.f. 10*x / ((1-x)*(x^2-98*x+1)). - R. J. Mathar, Dec 20 2011

a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>2. - Colin Barker, Mar 02 2016

a(n) = (-10+(5-2*sqrt(6))*(49+20*sqrt(6))^(-n)+(5+2*sqrt(6))*(49+20*sqrt(6))^n)/96. - Colin Barker, Mar 07 2016

EXAMPLE

3*0 = 2*0.

3*10 = 2*15.

3*990 = 2*1485.

3*97020 = 2*145530.

MATHEMATICA

LinearRecurrence[{99, -99, 1}, {0, 10, 990}, 20] (* Harvey P. Dale, Feb 25 2018 *)

PROG

(PARI) concat(0, Vec(10*x/((1-x)*(1-98*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 02 2016

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(10*x/((1-x)*(x^2-98*x+1)))); // G. C. Greubel, Jul 15 2018

CROSSREFS

Cf. A001652, A029549, A053141, A075528, A200994-A201008.

Sequence in context: A006242 A163566 A168520 * A062033 A171500 A154027

Adjacent sequences:  A200990 A200991 A200992 * A200994 A200995 A200996

KEYWORD

nonn,easy

AUTHOR

Charlie Marion, Dec 15 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 18 01:03 EDT 2021. Contains 347498 sequences. (Running on oeis4.)