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, ...

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

Offset: 1

Author

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

Keywords

• easy
• nonn

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 XA005408(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

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
• 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).

Programs

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

GAP

List([1..50], n-> n^2 +(n+1)^2); # G. C. Greubel, Sep 04 2019

Magma

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

Maple

seq(n^2 +(n+1)^2, n=1..50); # G. C. Greubel, Sep 04 2019

Mathematica

Table[n^2 +(n+1)^2, {n,50}]  (* Vladimir Joseph Stephan Orlovsky, Sep 25 2009, modified by G. C. Greubel, Sep 04 2019 *)
RecurrenceTable[{a[1]==5, a[n]==a[n-1] +4n}, a, {n, 50}] (* Vincenzo Librandi, Mar 03 2013 *)
LinearRecurrence[{3,-3,1},{5,13,25},50] (* Harvey P. Dale, Jul 16 2018 *)

PARI

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

Python

def A099776(n): return (n<<1)*(n+1)+1 # Chai Wah Wu, Oct 01 2024

Sage

[n^2 +(n+1)^2 for n in (1..50)] # G. C. Greubel, Sep 04 2019

Formula

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

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

From Colin Barker, Nov 03 2012: (Start)

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. (End)

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

E.g.f.: -1 +(1 +4*x +2*x^2)*exp(x). - G. C. Greubel, Sep 04 2019