cp's OEIS Frontend

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, ...

%I A099776 #55 Oct 02 2024 04:24:35
%S A099776 5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,
%T A099776 841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,2245,
%U A099776 2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325,4513
%N 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, ...
%C A099776 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
%C A099776 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
%C A099776 Sum of two consecutive squares: 1^4=5, 4+9=13, 9+16=25, 16+25=41, ... - _Vladimir Joseph Stephan Orlovsky_, Sep 25 2009
%C A099776 The sequence provides all integers m > 1 such that 2*m - 1 is a square. - _Vincenzo Librandi_, Mar 03 2013
%H A099776 Ray Chandler, <a href="/A099776/b099776.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi)
%H A099776 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A099776 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A099776 a(n) = ((2*n+1)^2 -1)/2 + 1.
%F A099776 a(n) = a(n-1) + 4*n for n>1, a(1)=5. - _Vincenzo Librandi_, Nov 17 2010
%F A099776 From _Colin Barker_, Nov 03 2012: (Start)
%F A099776 a(n) = 1 + 2*n + 2*n^2.
%F A099776 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A099776 G.f.: x*(5 -2*x +x^2)/(1-x)^3. (End)
%F A099776 All other formulas given in A001844 also apply, with the restriction n>0. - _M. F. Hasler_, Nov 03 2012
%F A099776 E.g.f.: -1 +(1 +4*x +2*x^2)*exp(x). - _G. C. Greubel_, Sep 04 2019
%p A099776 seq(n^2 +(n+1)^2, n=1..50); # _G. C. Greubel_, Sep 04 2019
%t A099776 Table[n^2 +(n+1)^2, {n,50}]  (* _Vladimir Joseph Stephan Orlovsky_, Sep 25 2009, modified by _G. C. Greubel_, Sep 04 2019 *)
%t A099776 RecurrenceTable[{a[1]==5, a[n]==a[n-1] +4n}, a, {n, 50}] (* _Vincenzo Librandi_, Mar 03 2013 *)
%t A099776 LinearRecurrence[{3,-3,1},{5,13,25},50] (* _Harvey P. Dale_, Jul 16 2018 *)
%o A099776 (C) #include "stdio.h"
%o A099776 int main(int argc, char* argv[]){
%o A099776   unsigned long i; int L = (argc>1) ? atol(argv[1]) : 50;
%o A099776   for (i=(L>0) ? 1 : (L*=-1); i<=L; i++)
%o A099776     printf ("%u, ", (i+1)*i*2+1);
%o A099776   return 0;
%o A099776 } // optional arg implemented by _M. F. Hasler_, Nov 03 2012
%o A099776 (Magma) [n eq 1 select 5 else Self(n-1)+4*n: n in [1..50]]; // _Vincenzo Librandi_, Mar 03 2013
%o A099776 (PARI) a(n)=1+2*n+2*n^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A099776 (Sage) [n^2 +(n+1)^2 for n in (1..50)] # _G. C. Greubel_, Sep 04 2019
%o A099776 (GAP) List([1..50], n-> n^2 +(n+1)^2); # _G. C. Greubel_, Sep 04 2019
%o A099776 (Python)
%o A099776 def A099776(n): return (n<<1)*(n+1)+1 # _Chai Wah Wu_, Oct 01 2024
%K A099776 easy,nonn
%O A099776 1,1
%A A099776 Nick Robins (nrobins(AT)hackettfreedman.com), Nov 12 2004