A105036 a(n) = 26*a(n-2) - a(n-4) + 12, with a(0) = 0, a(1) = 4, a(2) = 8, a(3) = 116.

0, 4, 8, 116, 220, 3024, 5724, 78520, 148616, 2038508, 3858304, 52922700, 100167300, 1373951704, 2600491508, 35669821616, 67512611920, 926041410324, 1752727418424, 24041406846820, 45503400267116, 624150536607008

Offset: 0

Author

Gerald McGarvey, Apr 03 2005

Keywords

• nonn

Comments

It appears this sequence gives all the nonnegative m such that 42*m^2 + 42*m + 1 is a square.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,26,-26,-1,1).

Crossrefs

Cf. A097309, A105037.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 4*x*(1+x+x^2)/((1-x)*(1-26*x^2+x^4)) )); // G. C. Greubel, Mar 15 2023

Mathematica

LinearRecurrence[{1,26,-26,-1,1},{0,4,8,116,220},30] (* Harvey P. Dale, Mar 25 2013 *)

SageMath

@CachedFunction
def a(n): # a = A105036
    if (n<5): return (0,4,8,116,220)[n]
    else: return a(n-1) +26*a(n-2) -26*a(n-3) -a(n-4) +a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Mar 15 2023

Formula

a(n) = 26*a(n-2) - a(n-4) + 12, for n > 3.

From R. J. Mathar, Sep 13 2009: (Start)

G.f.: 4*x*(1+x+x^2)/((1-x)*(1-26*x^2+x^4)).

a(n) = a(n-1) +26*a(n-2) -26*a(n-3) -a(n-4) +a(n-5). (End)

From Ralf Stephan, Nov 15 2010: (Start)

a(2n) = (1/2)*(A097309(n+2) - 9*A097309(n+1) - 1).

a(2n+1) = (1/2)*(9*A097309(n+2) - A097309(n+1) - 1). (End)