

A127838


a(1) = 1, a(2) = a(3) = a(4) = 0; a(n) = a(n4) + a(n3) for n > 4.


1



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
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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(nk) + a(nk+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 topleft element of the nth 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), 172184. [From N. J. A. Sloane, Dec 17 2010]
S. Suter, Binetlike formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007, apparently unpublished as of Mar 2014.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

Binetlike 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(n5) for n >= 5.
O.g.f.: x(x1)(1+x+x^2)/(x^4+x^31). (End)


MAPLE

P:=proc(n) local a, a0, a1, a2, a3, a4, i; a0:=1; a1:=0; a2:=0; a3:=0; print(a0); print(a1); print(a2); print(a3); for i from 1 by 1 to n do a:=a0+a1; a0:=a1; a1:=a2; a2:=a3; a3:=a; print(a); od; end: P(100); # Paolo P. Lava, Jun 28 2007


MATHEMATICA

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


CROSSREFS

Sequence in context: A247367 A305321 A340274 * A017817 A284834 A279677
Adjacent sequences: A127835 A127836 A127837 * A127839 A127840 A127841


KEYWORD

nonn,easy


AUTHOR

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


STATUS

approved



