OFFSET
0,2
COMMENTS
36*a(n) + 61 is a perfect square for even n > 0. 36*a(n) - 11 is a perfect square for odd n. - Klaus Purath, Apr 13 2026
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
G.f.: (1+2*x)*(1+2*x+2*x^2+2*x^3-x^4)/((1-x)^3*(1+x)^2). - Ralf Stephan, Apr 24 2004
From Colin Barker, Jan 26 2016: (Start)
a(n) = (18*n^2+2*(-1)^n*n+18*n-7*(-1)^n-1)/8 for n>0.
a(n) = (9*n^2+10*n-4)/4 for n>0 and even.
a(n) = (9*n^2+8*n+3)/4 for n odd. (End)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - Wesley Ivan Hurt, Apr 19 2023
From Klaus Purath, Apr 13 2026: (Start)
(a(2*n-1) + a(2*n))/18 = n^2.
a(n) = 2*a(n-2) - a(n-4) + 18 for n >= 5.
a(n) = a(n-2) + 9*n - 4 for even n >= 4.
a(n) = a(n-2) + 9*n - 5 for odd n. (End)
E.g.f.: (8 + (9*x^2 + 17*x - 4)*cosh(x) + (9*x^2 + 19*x + 3)*sinh(x))/4. - Stefano Spezia, May 03 2026
MATHEMATICA
Join[{1}, LinearRecurrence[{1, 2, -2, -1, 1}, {5, 13, 27, 45, 67}, 40]] (* Harvey P. Dale, Dec 03 2014 *)
a[0]=1; a[n_]:=If[EvenQ[n], 9*(n/2)^2+5*(n/2)-1, 9*((n-1)/2)^2+13*((n-1)/2)+5]; Table[a[n], {n, 0, 80}] (* Vincenzo Librandi, May 03 2026 *)
PROG
(PARI) a(n)=abs([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, -1, -2, 2, 1]^n*[-1; 5; 13; 27; 45])[1, 1] \\ Charles R Greathouse IV, Jun 16 2015
(PARI) Vec((1+2*x)*(1+2*x+2*x^2+2*x^3-x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016
(Magma) function a(n) if n eq 0 then return 1; elif n mod 2 eq 0 then m := n div 2; return 9*m^2+5*m-1; else m := (n-1) div 2; return 9*m^2+13*m+5; end if; end function; [a(n) : n in [0..80]]; // Vincenzo Librandi, May 03 2026
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved