

A117142


Number of partitions of n in which any two parts differ by at most 2.


13



1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 21, 27, 28, 35, 36, 44, 45, 54, 55, 65, 66, 77, 78, 90, 91, 104, 105, 119, 120, 135, 136, 152, 153, 170, 171, 189, 190, 209, 210, 230, 231, 252, 253, 275, 276, 299, 300, 324, 325, 350, 351, 377, 378, 405, 406, 434, 435, 464, 465
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Equals row sums of triangle A177991.  Gary W. Adamson, May 16 2010
Positive numbers that are either triangular (A000217) or triangular minus 1 (A000096).  Jon E. Schoenfield, Jun 12 2010


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jonathan Bloom, Nathan McNew, Counting patternavoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,2,2,1,1).


FORMULA

G.f.: Sum_{k>=1} x^k/((1  x^k)*(1  x^(k + 1))*(1  x^(k + 2))). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is Sum_{k>=1} (x^k/(Product_{j=k..k+b} 1  x^j)).
a(n) = (2*n^2 + 10*n + 3 + (1)^n * (2*n  3))/16.  Birkas Gyorgy, Feb 20 2011
G.f.: (1 + x)/(1  x)/(Q(0)  x^2  x^3), where Q(k) = 1 + x^2 + x^3 + k*x*(1 + x^2)  x^2*(1 + x*(k + 2))*(1 + k*x)/Q(k+1) ; (continued fraction).  Sergei N. Gladkovskii, Jan 05 2014
G.f.: x*(x^2  x  1) / ((x  1)^3*(x + 1)^2).  Colin Barker, Mar 05 2015
a(n) = Sum_{k=0..n1} A152271(k).  Jon Maiga, Dec 21 2018


EXAMPLE

a(6) = 9 because we have
1: [6],
2: [4,2],
3: [3,3],
4: [3,2,1],
5: [3,1,1,1],
6: [2,2,2],
7: [2,2,1,1],
8: [2,1,1,1,1], and
9: [1,1,1,1,1,1]
([5,1] and [4,1,1] do not qualify).


MAPLE

g:=sum(x^k/(1x^k)/(1x^(k+1))/(1x^(k+2)), k=1..75): gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..65); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]P[j][1]<3 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions


MATHEMATICA

Table[Count[IntegerPartitions[n], _?(Max[#]  Min[#] <= 2 &)], {n, 30}] (* Birkas Gyorgy, Feb 20 2011 *)
Table[(2 n^2 + 10 n + 3 + (1)^n (2 n  3))/16, {n, 30}] (* Birkas Gyorgy, Feb 20 2011 *)
Table[Sum[If[EvenQ[k], 1, (k + 1)/2], {k, 0, n}], {n, 0, 58}] (* Jon Maiga, Dec 21 2018 *)


PROG

(PARI) Vec(x*(x^2x1)/((x1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015
(GAP) List([1..60], n>(2*n^2+10*n+3+(1)^n*(2*n3))/16); # Muniru A Asiru, Dec 21 2018


CROSSREFS

Cf. A117143, A152271.
Cf. A177991.  Gary W. Adamson, May 16 2010
Cf. A000096, A000217.  Jon E. Schoenfield, Jun 12 2010
Column k=2 of A194621.  Alois P. Heinz, Oct 17 2012
Sequence in context: A290564 A167803 A092213 * A238617 A076061 A025523
Adjacent sequences: A117139 A117140 A117141 * A117143 A117144 A117145


KEYWORD

nonn,easy


AUTHOR

Emeric Deutsch, Feb 27 2006


STATUS

approved



