A152262 a(n) = 14*a(n-1) - 43*a(n-2), n > 1; a(0)=1, a(1)=7.

1, 7, 55, 469, 4201, 38647, 360415, 3383989, 31878001, 300780487, 2840172775, 26828857909, 253476581401, 2395031249527, 22630944493135, 213846879174229, 2020725695234401, 19094743928789767, 180435210107977495

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Keywords

• nonn
• easy

Comments

Binomial transform of A145301. Inverse binomial transform of A152263. - Philippe Deléham, Dec 03 2008

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
• Index entries for linear recurrences with constant coefficients, signature (14,-43).

Crossrefs

Cf. A027907, A098158, A145301, A152263.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((7+r6)^n+(7-r6)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

Magma

[n le 2 select 7^(n-1) else 14*Self(n-1) -43*Self(n-2): n in [1..30]]; // G. C. Greubel, May 23 2023

Mathematica

LinearRecurrence[{14,-43},{1,7},30] (* Harvey P. Dale, Apr 26 2015 *)

SageMath

@CachedFunction
def a(n): # a = A152262
    if (n<2): return 7^n
    else: return 14*a(n-1) - 43*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, May 23 2023

Formula

a(n) = ((7 + sqrt(6))^n + (7 - sqrt(6))^n)/2.

From Philippe Deléham, Dec 03 2008: (Start)

G.f.: (1-7*x)/(1-14*x+43*x^2).

a(n) = Sum_{k=0..n} A098158(n,k)*7^(2k-n)*6^(n-k). (End)

a(n) = Sum_{k=0..n} A027907(n,2k)*6^k. - J. Conrad, Aug 24 2016

E.g.f.: cosh(sqrt(6)*x)*exp(7*x). - Ilya Gutkovskiy, Aug 24 2016

a(n) = m^n*(ChebyshevU(n, 7/m) - (7/m)*ChebyshevU(n-1, 7/m)), with m = sqrt(43). - G. C. Greubel, May 23 2023

Extensions

Name from Philippe Deléham, Dec 03 2008