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A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8. 18
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Sometimes called the Fibonacci 9-step numbers.

For n >= 8, this gives the number of integers written without 0 in base ten, the sum of digits of which is equal to n-7. E.g., a(11) = 8 because we have the 8 numbers: 4, 13, 22, 31, 112, 121, 211, 1111.

The offset for this sequence is fairly arbitrary. - N. J. A. Sloane, Feb 27 2009

LINKS

T. D. Noe, Table of n, a(n) for n = 0..208

Martin Burtscher, Igor Szczyrba, RafaƂ Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

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

FORMULA

a(n) = Sum_{k=1..9} a(n-k) for n > 8, a(8) = 1, a(n) = 0 for n=0..7.

G.f.: x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9). - N. J. A. Sloane, Dec 04 2011

Another form of the g.f. f: f(z)=(z^8-z^9)/(1-2*z+z^(10)), then a(n)=sum((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i),i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i),i=0..floor((n-9)/10)) with sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

From N. J. A. Sloane, Dec 04 2011: (Start)

Let b be the smallest root (in magnitude) of g(x) := 1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9), b = 0.50049311828655225605926845999420216157202861343888...

Let c = -b^8/g'(b) = 0.00099310812055463178382193226558248643030626601288701...

Then a(n) is the nearest integer to c/b^n. (End)

MAPLE

for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-8-9*i, i)*2^(n-8-10*i), i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i, i)*2^(n-9-10*i), i=0..floor((n-9)/10)):od:seq(k(n), n=0..50); a:=taylor((z^8-z^9)/(1-2*z+z^(10)), z=0, 51); for p from 0 to 50 do j(p):=coeff(a, z, p):od :seq(j(p), p=0..50); # Richard Choulet, Feb 22 2010

MATHEMATICA

a={1, 0, 0, 0, 0, 0, 0, 0, 0}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]

LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

With[{nn=9}, LinearRecurrence[Table[1, {nn}], Join[Table[0, {nn-1}], {1}], 50]] (* Harvey P. Dale, Aug 17 2013 *)

PROG

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1; 1, 1, 1, 1, 1, 1, 1, 1, 1]^n*[0; 0; 0; 0; 0; 0; 0; 0; 1])[1, 1] \\ Charles R Greathouse IV, Jun 16 2015

(PARI) A104144(n, m=9)=(matrix(m, m, i, j, j==i+1||i==m)^n)[1, m] \\ M. F. Hasler, Apr 22 2018

CROSSREFS

Cf. A000045, A000073, A000078, A001591, A001592, A066178, A079262 (Fibonacci n-step numbers).

Cf. A255529 (Indices of primes in this sequence).

Sequence in context: A145115 A172318 A234590 * A258800 A194632 A282584

Adjacent sequences:  A104141 A104142 A104143 * A104145 A104146 A104147

KEYWORD

nonn,easy

AUTHOR

Jean Lefort (jlefort.apmep(AT)wanadoo.fr), Mar 07 2005

EXTENSIONS

Edited by N. J. A. Sloane, Aug 15 2006 and Nov 11 2006

Incorrect formula deleted by N. J. A. Sloane, Dec 04 2011

Name edited by M. F. Hasler, Apr 22 2018

STATUS

approved

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Last modified September 21 17:33 EDT 2018. Contains 315260 sequences. (Running on oeis4.)