|
| |
|
|
A001648
|
|
Tetranacci numbers A073817 without the leading term 4.
(Formerly M2648 N1055)
|
|
8
|
|
|
|
1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631, 5071, 9775, 18842, 36319, 70007, 134943, 260111, 501380, 966441, 1862875, 3590807, 6921503, 13341626, 25716811, 49570747, 95550687, 184179871, 355018116, 684319421, 1319068095, 2542585503, 4900991135
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
REFERENCES
|
Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=1..200
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Lucas n-Step Number
|
|
|
FORMULA
|
G.f.: x(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4).
a(n) = trace of M^n, where M = the 4 X 4 matrix [ 0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g. the trace (sum of diagonal terms) of M^12 = a(12) = 2631 = (108 + 316 + 717 + 1490). - Gary W. Adamson, Feb 22 2004
a(n)=n*sum(k=ceiling(n/5)..n, sum(i=0..(n-k)/4, (-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1))/k), n>0. [From Vladimir Kruchinin, Jan 20 2012]
|
|
|
MAPLE
|
A001648:=-(1+2*z+3*z**2+4*z**3)/(-1+z+z**2+z**3+z**4); [Conjectured by Simon Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
|
Rest@ CoefficientList[ Series[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, 40}], x] (* Or *)
a[0] = 4; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[n_] := 2*a[n - 1] - a[n - 5]; Array[a, 33] (* Robert G. Wilson v *)
LinearRecurrence[{1, 1, 1, 1}, {1, 3, 7, 15}, 60] (* From Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
|
|
|
PROG
|
(PARI) a(n)=if(n<0, 0, polcoeff(x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4)+x*O(x^n), n))
(Maxima) a(n):=n*sum(sum((-1)^i*binomial(k, k-i)*binomial(n-i*4-1, k-1), i, 0, ((n-k)/4))/k, k, ceiling(n/5), n); [From Vladimir Kruchinin, Jan 20 2012]
|
|
|
CROSSREFS
|
Cf. A000288, A073817.
Sequence in context: A078869 A011890 A131076 * A051054 A001649 A213215
Adjacent sequences: A001645 A001646 A001647 * A001649 A001650 A001651
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
