OFFSET
0,6
COMMENTS
This sequence is an example of a "symmetric" quartic recurrence and has some expected divisibility properties.
a(n-3) counts partially ordered partitions of (n-3) into parts 1,2,3 where only the order of the adjacent 1's and 3's are unimportant (see example). - David Neil McGrath, Jul 25 2015
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..4239
Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1).
FORMULA
G.f.: x^3/(x^4 - x^3 - x^2 - x + 1).
a(n) = -a(2-n) for all n in Z. - Michael Somos, Jul 25 2025
EXAMPLE
Partially ordered partitions of (n-3) into parts 1,2,3 where only the order of adjacent 1's and 3's are unimportant. E.g., a(n-3)=a(6)=19. These are (33),(321),(312),(231),(123),(132),(3111),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111). - David Neil McGrath, Jul 25 2015
G.f. = x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 11*x^8 + 19*x^9 + 32*x^10 + ... - Michael Somos, Jul 25 2025
MATHEMATICA
LinearRecurrence[{1, 1, 1, -1}, {0, 0, 0, 1}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
CoefficientList[Series[x^3/(1-x-x^2-x^3+x^4), {x, 0, 40}], x] (* Harvey P. Dale, Mar 25 2018 *)
PROG
(PARI) v=[0, 0, 0, 1]; for(i=1, 40, v=concat(v, v[#v]+v[#v-1]+v[#v-2]-v[#v-3])); v \\ Derek Orr, Aug 27 2015
(PARI) {a(n) = if(n<0, -a(2-n), polcoeff(x^3/(1 - x - x^2 - x^3 + x^4 + x*O(x^n)), n))} /* Michael Somos, Jul 25 2025 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. K. Guy, Mar 23 2008
EXTENSIONS
More terms from Max Alekseyev, Mar 23 2008
Name clarified by Michael Somos, Jul 25 2025
STATUS
approved
