A117142 Number of partitions of n in which any two parts differ by at most 2.

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

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 and Nathan McNew, Counting pattern-avoiding 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*(1 + x - x^2)/((1 - x)^3*(1 + x)^2). - Colin Barker, Mar 05 2015

a(n) = Sum_{k=0..n-1} A152271(k). - Jon Maiga, Dec 21 2018

E.g.f.: (1/16)*( (3 + 2*x)*exp(-x) + (3 + 12*x + 2*x^2)*exp(x) ). - G. C. Greubel, Jul 18 2023

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],
  9: [1, 1, 1, 1, 1, 1]
([5,1] and [4,1,1] do not qualify).

Maple

g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(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, 60}] (* Jon Maiga, Dec 21 2018 *)

Prog

(PARI) Vec(x*(x^2-x-1)/((x-1)^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*n-3))/16); # Muniru A Asiru, Dec 21 2018
(Magma) [(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16: n in [1..60]]; // G. C. Greubel, Jul 18 2023
(SageMath) [(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16 for n in range(1, 61)] # G. C. Greubel, Jul 18 2023

Crossrefs

Cf. A000096, A000217, A117143, A152271, A177991.

Column k=2 of A194621. - Alois P. Heinz, Oct 17 2012

Sequence in context: A167803 A367402 A092213 * A238617 A076061 A025523

Adjacent sequences: A117139 A117140 A117141 * A117143 A117144 A117145

Keyword

nonn,easy

Author

Emeric Deutsch, Feb 27 2006

Status

approved