login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099776 Length of the hypotenuse of an integer right triangle with the hypotenuse being one more than the longer side. The shorter sides are just consecutive odd numbers 3, 5, 7, ... 7
5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Largest hypotenuse of primitive Pythagorean triangles with inradius n. (For smallest hypotenuse of PPT with inradius n, see A087484.)  Essentially the same as A001844. - Lekraj Beedassy, May 08 2006

The complete triple {X(n), Y(n), Z(n)=Y(n)+1}, with X<Y<Z, {X(n)=A005408(n);Y(n)=A046092(n), Z(n)=A001844(n)} may be recursively generated through the mapping W(n) -> M*W(n), where W(n) = transpose of vector [X(n) Y(n) Z(n)] and M a 3 X 3 matrix given by [1 -2 2 / 2 -1 2 / 2 -2 3 ]. Such triples correspond to successive number pair Pythagorean generators(p,q=p+1) yielding {X=p+q,Y=2p*q,Z=p^2 + q^2}. - Lekraj Beedassy, Jun 04 2006

Sum of two consecutive squares: 1^4=5, 4+9=13, 9+16=25, 16+25=41, ... - Vladimir Joseph Stephan Orlovsky, Sep 25 2009

The sequence provides all integers m > 1 such that 2*m - 1 is a square. - Vincenzo Librandi, Mar 03 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = ((2*n+1)^2-1)/2 + 1.

a(n) = 4*n+a(n-1) for n>1, a(1)=5. - Vincenzo Librandi, Nov 17 2010

a(n) = 1+2*n+2*n^2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(5-2*x+x^2)/(1-x)^3. - Colin Barker, Nov 03 2012

All other formulas given in A001844 also apply, with the restriction n>0. - M. F. Hasler, Nov 03 2012

MATHEMATICA

lst={}; Do[a=(n^2+(n+1)^2); AppendTo[lst, a], {n, 5!}]; lst  (* Vladimir Joseph Stephan Orlovsky, Sep 25 2009 *)

RecurrenceTable[{a[1]==5, a[n]==a[n-1] + 4 n}, a, {n, 50}] (* Vincenzo Librandi, Mar 03 2013 *)

LinearRecurrence[{3, -3, 1}, {5, 13, 25}, 50] (* Harvey P. Dale, Jul 16 2018 *)

PROG

(C) #include "stdio.h"

int main(int argc, char* argv[]){

  unsigned long i; int L = (argc>1) ? atol(argv[1]) : 50;

  for (i=(L>0) ? 1 : (L*=-1); i<=L; i++)

    printf ("%u, ", (i+1)*i*2+1);

  return 0;

} // optional arg implemented by M. F. Hasler, Nov 03 2012

(MAGMA) [n eq 1 select 5 else Self(n-1)+4*n: n in [1..50]]; // Vincenzo Librandi, Mar 03 2013

(PARI) a(n)=1+2*n+2*n^2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Sequence in context: A081961 A096891 A001844 * A301302 A133322 A299258

Adjacent sequences:  A099773 A099774 A099775 * A099777 A099778 A099779

KEYWORD

easy,nonn

AUTHOR

Nick Robins (nrobins(AT)hackettfreedman.com), Nov 12 2004

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 July 23 19:33 EDT 2019. Contains 325263 sequences. (Running on oeis4.)