A122074 a(0)=1, a(1)=6, a(n) = 7*a(n-1) - 2*a(n-2).

1, 6, 40, 268, 1796, 12036, 80660, 540548, 3622516, 24276516, 162690580, 1090281028, 7306586036, 48965540196, 328145609300, 2199088184708, 14737326074356, 98763106151076, 661867090908820, 4435543424059588, 29725069786599476, 199204401658077156

Offset: 0

Author

Gary W. Adamson, Oct 16 2006

Keywords

• nonn
• easy

Comments

First row sum of the matrix M^n, where M is the 3 X 3 matrix {{2,2,2},{2,3,2},{2,2,3}}.

Examples

			a(2)=40 because M^2={{12,14,14},{14,17,16},{14,16,17}} and 12+14+14=40.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..200
• J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 18.
• Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
• Index entries for linear recurrences with constant coefficients, signature (7,-2).

Programs

GAP

a:=[1,6];; for n in [3..30] do a[n]:=7*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 02 2019

Magma

I:=[1,6]; [n le 2 select I[n] else 7*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 03 2015

Maple

with(linalg): M[1]:=matrix(3,3,[2,2,2,2,3,2,2,2,3]): for n from 2 to 20 do M[n]:=multiply(M[n-1],M[1]) od: 1,seq(M[n][1,1]+M[n][1,2]+M[n][1,3],n=1..20);
# alternative:
f:= gfun:-rectoproc({a(n+2)-7*a(n+1)+2*a(n),a(0)=1,a(1)=6},a(n),remember):
seq(f(n),n=0..30); # Robert Israel, Oct 02 2015

Mathematica

M = {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
Transpose[NestList[{Last[#],7*Last[#]-2*First[#]}&,{1,6},25]] [[1]] (* Harvey P. Dale, Mar 11 2011 *)
f[s_] := Append[s, 7*s[[-1]] - 2*s[[-2]]]; Nest[f, {1, 6}, 18] (* Robert G. Wilson v, Mar 12 2011 *)
LinearRecurrence[{7,-2},{1,6},25] (* Harvey P. Dale, Jan 04 2014 *)

PARI

Vec((1-x)/(1-7*x+2*x^2) + O(x^30)) \\ Michel Marcus, Oct 03 2015

Sage

def A122074_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-7*x+2*x^2)).list()
A122074_list(30) # G. C. Greubel, Oct 02 2019

Formula

a(n) = 8*a(n-1) - 9*a(n-2) + 2*a(n-3); a(0)=1, a(1)=6, a(2)=40 (follows from the minimal polynomial x^3 - 8x^2 + 9x - 2 of M).

a(n) = (1/2 + 5*sqrt(41)/82)*(7/2 + sqrt(41)/2)^n + (1/2 - 5*sqrt(41)/82)*(7/2 - sqrt(41)/2)^n. - Antonio Alberto Olivares, Jun 06 2011

G.f.: (1-x)/(1-7*x+2*x^2). - Colin Barker, Feb 08 2012

E.g.f.: (1/41)*exp(7*x/2)*(41*cosh(sqrt(41)*x/2) + 5*sqrt(41)*sinh(sqrt(41)*x/2)). - Stefano Spezia, Oct 03 2019

Extensions

Edited by N. J. A. Sloane, Oct 29 2006 and Dec 04 2006