A256278 a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).

1, 2, 33, 965, 28958, 869713, 26121321, 784539274, 23563199185, 707707535789, 21255600833094, 638400107288033, 19173990901769297, 575880114843495250, 17296237823997043137, 519482849213446974997, 15602377428720941973934, 468608697663159238917041

Offset: 0

Author

Karl V. Keller, Jr., Jun 02 2015

Keywords

• nonn
• easy
• changed

Comments

The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).

The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29))) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).

Examples

			For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.
		

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Golden Ratio
• Eric Weisstein's World of Mathematics, Golden Ratio Conjugate
• Index entries for linear recurrences with constant coefficients, signature (31,-29).

Crossrefs

Cf. A001622, A195819 (29*n), A084330, A004922, A004942, A004962, A248786, A249079, A248739.

Programs

Magma

I:=[1,2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 03 2015

Maple

a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1,1]:
seq(a(n), n=0..23);  # Alois P. Heinz, Dec 22 2023

Mathematica

LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 03 2015 *)

Python

print(1, end=', ')
print(2, end=', ')
an = [1,2]
for n in range(2,26):
  print(31*an[n-1]-29*an[n-2], end=', ')
  an.append(31*an[n-1]-29*an[n-2])

Formula

G.f.: (1-29*x)/(29*x^2-31*x+1). - Vincenzo Librandi, Jun 03 2015

E.g.f.: exp(31*x/2)*(65*cosh(13*sqrt(5)*x/2) - 27*sqrt(5)*sinh(13*sqrt(5)*x/2))/65. - Stefano Spezia, Aug 31 2025