A086901 a(1) = a(2) = 1; for n>2, a(n) = 4*a(n-1) + 3*a(n-2).

1, 1, 7, 31, 145, 673, 3127, 14527, 67489, 313537, 1456615, 6767071, 31438129, 146053729, 678529303, 3152278399, 14644701505, 68035641217, 316076669383, 1468413601183, 6821884412881, 31692778455073, 147236767058935

Offset: 1

Author

Rick Powers (rick.powers(AT)mnsu.edu), Sep 18 2003

Keywords

• easy
• nonn

Examples

			a(3) = 4*1 + 3*1 = 7;
a(4) = 4*7 + 3*1 = 31.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
• Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.

Crossrefs

Cf. A015530, A102900, A122542.

Programs

Haskell

a086901 n = a086901_list !! (n-1)
a086901_list = 1 : 1 : zipWith (+)
               (map (* 3) a086901_list) (map (* 4) $ tail a086901_list)
-- Reinhard Zumkeller, Feb 13 2015

Magma

[n le 2 select 1 else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 28 2024

Mathematica

a[n_]:=(MatrixPower[{{3,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,4},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{4,3}, {1,1}, 41] (* G. C. Greubel, Oct 28 2024 *)

PARI

A086901(n)=if(n<3,1,4*A086901(n-1)+3*A086901(n-2)) \\ Michael B. Porter, Apr 04 2010

SageMath

A086901=BinaryRecurrenceSequence(4,3,1,1)
[A086901(n) for n in range(41)] # G. C. Greubel, Oct 28 2024

Formula

a(n) = ((c + 5)*b^n - (b + 5)*c^n)/14, where b = 2 + sqrt(7), c = 2 - sqrt(7).

From Ralf Stephan, Feb 01 2004: (Start)

G.f.: x*(1-3*x)/(1 - 4*x - 3*x^2).

a(n) = A015530(n) - 3*A015530(n-1) = 1 + 6*Sum_{k=0..n-2} A015530(k). (End)

a(n+1) = Sum_{k=0..n} 3^(n-k)*A122542(n,k), n>=0. - Philippe Deléham, Oct 27 2006

a(n) = upper left term in the 2 X 2 matrix [1,2; 3,3]^(n-1). - Gary W. Adamson, Mar 02 2008

G.f.: G(0)*(1-3*x)/(2-4*x), where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

E.g.f.: exp(2*x)*( cosh(sqrt(7)*x) - (1/sqrt(7))*sinh(sqrt(7)*x) ). - G. C. Greubel, Oct 28 2024

Extensions

More terms from Ray Chandler, Sep 19 2003