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A062544 a(n) = n plus sum of previous three terms. 6
0, 1, 3, 7, 15, 30, 58, 110, 206, 383, 709, 1309, 2413, 4444, 8180, 15052, 27692, 50941, 93703, 172355, 317019, 583098, 1072494, 1972634, 3628250, 6673403, 12274313, 22575993, 41523737, 76374072, 140473832, 258371672, 475219608 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It appears that this is the number of nonempty subsets of {1,2,...,n} with no gap of length greater than 3 (a set S has a gap of length d if a and b are in S but no x with a<x<b is in S, where b-a=d). See A119407 for the corresponding problem for gaps of length 4. - John W. Layman, Nov 02 2011

a(n-3) is the number of compositions of n with no part divisible by 3 and an odd number of parts congruent to 4 or 5 modulo 6. See Moser & Whitney reference. a(2) = 3 counts (5), (4,1), and (1,4) among the compositions of 5. - Brian Hopkins, Sep 06 2019

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..300

L. Moser and E. L. Whitney, Weighted compositions, Canad. Math. Bull. 4 (1961), 39-43.

Index entries for linear recurrences with constant coefficients, signature (3, -2, 0, -1, 1).

FORMULA

a(n) = 3*a(n-1) - 2*a(n-2) - 1*a(n-4) + 1*a(n-5). - Joerg Arndt, Apr 02 2011

a(n) = n + a(n-1) + a(n-2) + a(n-3) =(A001590(n+4) - n - 3)/2.

G.f.: x / ((1 - x) * (1 - 2*x + x^4)). a(n) = 2*a(n-1) - a(n-4) + 1. - Michael Somos, Dec 28 2012

a(n) = A325473(n+3) - (n+3). - Brian Hopkins, Sep 06 2019

EXAMPLE

a(5) = 5 + 15 + 7 + 3 = 30.

x + 3*x^2 + 7*x^3 + 15*x^4 + 30*x^5 + 58*x^6 + 110*x^7 + 206*x^8 + 383*x^9 + ...

MATHEMATICA

Join[{c=0}, a=b=0; Table[z=b+a+c+n; a=b; b=c; c=z, {n, 1, 40}]] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)

PROG

(PARI) { a=a1=a2=a3=0; for (n=0, 300, write("b062544.txt", n, " ", a+=n + a2 + a3); a3=a2; a2=a1; a1=a ) } \\ Harry J. Smith, Aug 08 2009

(PARI) {a(n) = if( n<0, n = -n; polcoeff( x^4 / ((1 - x) * (1 - 2*x^3 + x^4)) + x * O(x^n), n), polcoeff( x / ((1 - x) * (1 - 2*x + x^4)) + x * O(x^n), n))} /* Michael Somos, Dec 28 2012 */

CROSSREFS

n plus sum of all previous terms gives A000225, n plus sum of two previous terms gives A001924, n plus previous term gives A000217, n gives A001477.

Cf. A007800, A119407.

Cf. A001590 and A325473.

Sequence in context: A209816 A182726 A023610 * A120411 A224520 A291754

Adjacent sequences:  A062541 A062542 A062543 * A062545 A062546 A062547

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jun 26 2001

STATUS

approved

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)