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
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).
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
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 *)
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
(Sage) [n^2 +(n+1)^2 for n in (1..50)] # G. C. Greubel, Sep 04 2019
(GAP) List([1..50], n-> n^2 +(n+1)^2); # G. C. Greubel, Sep 04 2019
(Python)
def A099776(n): return (n<<1)*(n+1)+1 # Chai Wah Wu, Oct 01 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Nick Robins (nrobins(AT)hackettfreedman.com), Nov 12 2004
STATUS
approved