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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 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*(x^2 - x - 1) / ((x - 1)^3*(x + 1)^2). - Colin Barker, Mar 05 2015

a(n) = Sum_{k=0..n-1} 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/(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, 58}] (* 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

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

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Last modified December 8 08:49 EST 2019. Contains 329862 sequences. (Running on oeis4.)