A127838 a(1) = 1, a(2) = a(3) = a(4) = 0; a(n) = a(n-4) + a(n-3) for n > 4.

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 4, 6, 5, 6, 10, 11, 11, 16, 21, 22, 27, 37, 43, 49, 64, 80, 92, 113, 144, 172, 205, 257, 316, 377, 462, 573, 693, 839, 1035, 1266, 1532, 1874, 2301, 2798, 3406, 4175, 5099, 6204, 7581

Offset

1,12

Comments

Part of the phi_k family of sequences defined by a(1)=1, a(2)=...=a(k)=0, a(n) = a(n-k) + a(n-k+1) for n > k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.

The sequence can be interpreted as the top-left element of the n-th power of 6 different 4 X 4 (0,1) matrices. - R. J. Mathar, Mar 19 2014

References

G. Mantel, Resten van wederkeerige Reeksen (Remainders of the reciprocal series), Nieuw Archief v. Wiskunde, 2nd series, I (1894), 172-184. [From N. J. A. Sloane, Dec 17 2010]

S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007, apparently unpublished as of Mar 2014.

Links

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).

Formula

Binet-like formula: a(n) = Sum_{i=1...4} (r_i^n)/(3(r_i)^2+4(r_i)) where r_i is a root of x^4=x+1.

From R. J. Mathar, Mar 06 2008: (Start)

a(n) = A017817(n-5) for n >= 5.

O.g.f.: x(x-1)(1+x+x^2)/(x^4+x^3-1). (End)

Mathematica

LinearRecurrence[{0, 0, 1, 1}, {1, 0, 0, 0}, 60] (* Harvey P. Dale, Feb 15 2015 *)

Crossrefs

Sequence in context: A305321 A346798 A340274 * A017817 A363567 A284834

Adjacent sequences: A127835 A127836 A127837 * A127839 A127840 A127841

Keyword

nonn,easy

Author

Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007

Status

approved