A088585 -id:A088585 - cp's OEIS frontend

A121238 a(n) = (-1)^(1+n+A088585(n)).

1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1

Offset: 0

Philippe Deléham, Aug 21 2006

sign

Apparently the partial products of this sequence form the Hankel transform of A023359: 1, 1*1 = 1, 1*1*1 = 1, 1*1*1*-1 = -1, 1*1*1*-1*1 = -1, ... and 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, ... is the Hankel transform of A023359.

Index entries for sequences obtained by enumerating foldings

Cf. A023359, A088585.

A088567 Number of "non-squashing" partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19, 24, 25, 31, 32, 40, 41, 50, 51, 63, 64, 77, 78, 95, 96, 114, 115, 138, 139, 163, 164, 194, 195, 226, 227, 266, 267, 307, 308, 357, 358, 408, 409, 471, 472, 535, 536, 612, 613, 690, 691, 785, 786, 881, 882, 995, 996, 1110, 1111

Offset: 0

N. J. A. Sloane, Nov 30 2003

nonn

"Non-squashing" refers to the property that p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k: if the parts are stacked in increasing size, at no point does the sum of the parts above a certain part exceed the size of that part.

			The partitions of n = 1 through 6 are: 1; 2; 3=1+2; 4=1+3; 5=1+4=2+3; 6=1+5=2+4=1+2+3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Amanda Folsom et al., On a general class of non-squashing partitions, Discrete Mathematics 339.5 (2016): 1482-1506.
Y. Homma, J. H. Ryu, and B. Tong, Sequence non-squashing partitions, Slides from a talk, 2014.
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Cf. A000123, A088575, A088585, A088931, A089054. A090678 gives sequence mod 2.

Cf. A187821 (non-squashing partitions of n into odd parts).

import Data.List (transpose)
a088567 n = a088567_list !! n
a088567_list = 1 : tail xs where
   xs = 0 : 1 : zipWith (+) xs (tail $ concat $ transpose [xs, tail xs])
-- Reinhard Zumkeller, Nov 15 2012

f := proc(n) option remember; local t1,i; if n <= 2 then RETURN(1); fi; t1 := add(f(i),i=0..floor(n/2)); if n mod 2 = 0 then RETURN(t1-1); fi; t1; end;
t1 := 1 + x/(1-x); t2 := add( x^(3*2^(k-1))/ mul( (1-x^(2^j)),j=0..k), k=1..10); series(t1+t2, x, 256); # increase 10 to get more terms

max = 63; f = 1 + x/(1-x) + Sum[x^(3*2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014 *)

a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(2m-1) + a(m) - 1, a(2m+1) = a(2m) + 1.
Or, a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(0)+a(1)+...+a(m)-1; a(2m+1) = a(0)+a(1)+...+a(m).
G.f.: 1 + x/(1-x) + Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1-x^(2^j)).
G.f.: Product_{n>=0} 1/(1-x^(2^n)) - Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ). (The two terms correspond to A000123 and A088931 respectively.)

A088575 Bisection of A088567.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 13, 18, 24, 31, 40, 50, 63, 77, 95, 114, 138, 163, 194, 226, 266, 307, 357, 408, 471, 535, 612, 690, 785, 881, 995, 1110, 1248, 1387, 1550, 1714, 1908, 2103, 2329, 2556, 2822, 3089, 3396, 3704, 4061, 4419, 4827, 5236, 5707, 6179, 6714, 7250, 7862

Offset: 0

N. J. A. Sloane, Nov 30 2003

nonn

(1 + x^2 + x^3 + x^4 + x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + ...) = (1 + x + 2x^2 + 4x^3 + 6x^4 + ...). - Gary W. Adamson, Jul 27 2010

Cf. A088567, A088585.

a(n) = A088585(n) - 1 for n >= 1.

a(n) = a(n-1) + a(floor(n/2)) + (1-(-1)^n)/2, n > 1. - Vladeta Jovovic, Dec 02 2003

A242634 G.f. A(x) satisfies A(x) = A(x^2) / (1 - x) + x / (1 - x^2).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19, 24, 25, 31, 32, 40, 41, 50, 51, 63, 64, 77, 78, 95, 96, 114, 115, 138, 139, 163, 164, 194, 195, 226, 227, 266, 267, 307, 308, 357, 358, 408, 409, 471, 472, 535, 536, 612, 613, 690, 691, 785, 786, 881, 882

Offset: 0

Michael Somos, May 19 2014

nonn

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A088567, A088585.

a[ n_] := If[ n < 0, 0, Module[{A = 0}, Do[A = (x + (1 + x) (A /. x -> x^2)) / (1 - x^2), {IntegerLength[ n, 2]}]; SeriesCoefficient[ A, {x, 0, n}]]];

{a(n) = my(A = O(x)); if( n<0, 0, for(k=1, #binary(n), A = (x + (1 + x) * subst(A, x, x^2)) / (1 - x^2)); polcoeff(A, n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, #binary(n\3), x^(2^k*3 \ 2) / prod(j=0, k, 1 - x^2^j), x * O(x^n)), n))};

G.f.: x / (1 - x) + Sum_{k>0} x^(3*2^(k-1)) / Product_{j=0..k} (1 - x^(2^j)).
a(n) = a(n-2) + a(floor(n/2)) unless n=1.
a(n) = A088585(n) - A088585(n-1) if n>=1.
a(n) = A088567(n) if n>0.
a(2*n + 1) = a(2*n) + 1 = A088585(n) if n>=0.

cp's OEIS Frontend

A121238 a(n) = (-1)^(1+n+A088585(n)).

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A088567 Number of "non-squashing" partitions of n into distinct parts.

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A088575 Bisection of A088567.

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A242634 G.f. A(x) satisfies A(x) = A(x^2) / (1 - x) + x / (1 - x^2).

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