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A279196 Number of polynomials of the form P(x,y) = 1 + (x+y-1) * Q(x,y) such that P(1,1) = n and both polynomials P and Q have nonnegative integer coefficients. 2
1, 1, 2, 5, 13, 36, 102, 295, 864, 2557, 7624, 22868, 68920, 208527, 632987, 1926752, 5878738, 17973523, 55050690, 168881464, 518818523, 1595878573, 4914522147, 15150038699, 46747391412, 144370209690, 446214862158, 1380161749537, 4271808447154, 13230257155092, 40999697820032 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Original definition did not have the requirement for Q to have nonnegative coefficients. However, this results in different terms given by A363933. We have a(n) <= A363933(n), which is strict for n >= 5. - Max Alekseyev, Jun 28 2023
In colorful terms, one can view a polynomial as a configuration made of piles of tokens located at lattice points (i>=0, j>=0). One introduces the notion of "degradation of a configuration": to degrade a configuration, choose a nonempty pile of tokens in it, say that at (i,j); remove one token from that pile; then add one token at (i+1,j) and one token at (i,j+1). This is a nondeterministic process. a(n) is the number of distinct configurations one can possibly get after (n-1) degradations of the initial configuration consisting of just one token at (0,0). In this metaphor, the P's are the resulting configurations and the Q's are records of where the tokens have been taken. - Luc Rousseau, Jun 30 2023
LINKS
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "D".
N. J. A. Sloane, Winter Fruits: New Problems from OEIS, Dec. 2016 - Jan. 2017 (part 1), 2017-01-26, (discussion from 6:23-10:00).
EXAMPLE
From Peter Kagey, Feb 03 2017 (Start):
For n = 1 the a(1) = 1 solution is:
1 = 0(x + y - 1) + 1.
For n = 2 the a(2) = 1 solution is:
x + y = (x + y - 1) + 1.
For n = 3 the a(3) = 2 solutions are:
xy + x + y^2 = (y + 1)(x + y - 1) + 1;
xy + y + x^2 = (x + 1)(x + y - 1) + 1.
For n = 4 the a(4) = 5 solutions are:
x^2 + 2xy + y^2 = (x + y + 1)(x + y - 1) + 1;
x^2y + x^2 + xy^2 + y = (xy + x + 1)(x + y - 1) + 1;
x^2y + xy^2 + x + y^2 = (xy + y + 1)(x + y - 1) + 1;
xy^2 + xy + x + y^3 = (y^2 + y + 1)(x + y - 1) + 1;
x^3 + x^2y + xy + y = (x^2 + x + 1)(x + y - 1) + 1.
(End) [Corrected by Luc Rousseau, Jun 30 2023]
PROG
(Java) // See link. Luc Rousseau, Jun 30 2023
(Java + Prolog) // See link. Luc Rousseau, Jun 30 2023
(Prolog) % See link. Luc Rousseau, Jul 26 2023
CROSSREFS
Cf. A363933.
Sequence in context: A356697 A346738 A116409 * A002844 A223096 A277996
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 15 2016
EXTENSIONS
Definition corrected by Max Alekseyev, Jun 28 2023
a(10)-a(18) from Luc Rousseau, Jun 30 2023
a(19)-a(25) from Max Alekseyev, Jul 04 2023
a(26)-a(29) from Luc Rousseau, Jul 31 2023
a(30) from Luc Rousseau, Nov 10 2023
a(31) from Luc Rousseau, Dec 18 2023
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)