A357628 -id:A357628 - cp's OEIS frontend

A357624 Skew-alternating sum of the reversed n-th composition in standard order.

0, 1, 2, 0, 3, -1, 1, -1, 4, -2, 0, -2, 2, -2, 0, 0, 5, -3, -1, -3, 1, -3, -1, 1, 3, -3, -1, -1, 1, -1, 1, 1, 6, -4, -2, -4, 0, -4, -2, 2, 2, -4, -2, 0, 0, 0, 2, 2, 4, -4, -2, -2, 0, -2, 0, 2, 2, -2, 0, 0, 2, 0, 2, 0, 7, -5, -3, -5, -1, -5, -3, 3, 1, -5, -3, 1

Offset: 0

Gus Wiseman, Oct 08 2022

sign

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 357-th composition is (2,1,3,2,1) so a(357) = 1 - 2 - 3 + 2 + 1 = -1.
The 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 1 + 2 = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The half-alternating form is A357622, non-reverse A357621.
The reverse version is A357623.
Positions of zeros are A357628, non-reverse A357627.
The version for prime indices is A357630.
The version for Heinz numbers of partitions is A357634.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001700, A001511, A053251, A344619, A357136, A357182, A357183, A357184, A357185, A357625, A357626, A357629, A357640.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[skats[Reverse[stc[n]]],{n,0,100}]

A228248 Number of 2n-step lattice paths from (0,0) to (0,0) using steps in {N, S, E, W} starting with East, then always moving straight ahead or turning left.

Original entry on oeis.org

1, 0, 1, 3, 9, 30, 103, 357, 1257, 4494, 16246, 59246, 217719, 805389, 2996113, 11200113, 42047593, 158452138, 599113966, 2272065638, 8639763574, 32933685102, 125817012366, 481631387438, 1847110931703, 7095928565405, 27302745922817, 105204285608025

Offset: 0

David Scambler and Alois P. Heinz, Aug 18 2013

From Gus Wiseman, Oct 13 2022: (Start)
Also the number of integer compositions of 2n whose half-alternating and skew-alternating sums are both 0, where we define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ..., and the skew-alternating sum to be A - B - C + D + E - F - G + ... For example, the a(0) = 1 through a(4) = 9 compositions are:
  ()  .  (1111)  (1212)   (1313)
                 (2121)   (2222)
                 (11211)  (3131)
                          (11312)
                          (12221)
                          (21311)
                          (112211)
                          (1112111)
                          (11111111)
For skew-alternating only: A001700, ranked by A357627, reverse A357628.
For partitions: A035363, half only A357639, skew only A357640.
For half-alternating only: A357641, ranked by A357625, reverse A357626.
These compositions are ranked by A357706.
(End)

			a(0) = 1: [], the empty path.
a(1) = 0.
a(2) = 1: ENWS.
a(3) = 3: EENWWS, ENNWSS, ENWWSE.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
David Scambler et al., A new lattice walk and follow-up messages on the SeqFan list, May 08 2013.

Cf. A004006 (same rules, but self-avoiding).

Cf. A000583, A001511, A001700, A088218, A357136, A357182, A357642.

b:= proc(x, y, n) option remember; `if`(abs(x)+abs(y)>n, 0,
      `if`(n=0, 1, b(x+1, y, n-1) +b(y+1, -x, n-1)))
    end:
a:= n-> ceil(b(0, 0, 2*n)/2):
seq(a(n), n=0..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 0, 1, 3, 9][n+1],
     ((n-1)*(414288-1901580*n+186029*n^6-869551*n^5+2393807*n^4
     -3938624*n^3+3753546*n^2+1050*n^8-21605*n^7)*a(n-1)
     +(-17751540*n-12215020*n^5+3494038*n^6+3777840+27478070*n^4
     -39711374*n^3+35488098*n^2-2700*n^9+62370*n^8-621126*n^7)*a(n-2)
     +(-18193248+77490792*n-9138800*n^6+35323128*n^5-88122332*n^4
     +141370392*n^3-140075264*n^2+5400*n^9-135540*n^8+1476432*n^7)*a(n-3)
     +(-192473328*n+48577536+17091500*n^6-70036368*n^5+184890672*n^4
     -313388816*n^3+328043052*n^2-8400*n^9+224440*n^8-2600032*n^7)*a(n-4)
     +8*(n-5)*(150*n^6-2015*n^5+10852*n^4-29867*n^3+44208*n^2-33540*n
     +10416)*(-9+2*n)^2*a(n-5)) / (n^2*(396988*n-487261*n^2+150*n^7
     -3065*n^6+26092*n^5-119602*n^4+317746*n^3-131048)))
    end:
seq(a(n), n=0..40);

b[x_, y_, n_] := b[x, y, n] = If[Abs[x] + Abs[y] > n, 0, If[n == 0, 1, b[x + 1, y, n - 1] + b[y + 1, -x, n - 1]]];
a[n_] := Ceiling[b[0, 0, 2n]/2];
a /@ Range[0, 40] (* Jean-François Alcover, May 14 2020, after Maple *)
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n],halfats[#]==0&&skats[#]==0&]],{n,0,7}] (* Gus Wiseman, Oct 12 2022 *)

a(n) ~ 2^(2n-1)/(Pi*n). - Vaclav Kotesovec, Jul 16 2014

A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

Original entry on oeis.org

0, 15, 45, 54, 59, 153, 170, 179, 204, 213, 230, 235, 247, 255, 561, 594, 611, 660, 677, 710, 715, 727, 735, 750, 765, 792, 809, 842, 851, 871, 879, 894, 908, 917, 934, 939, 951, 959, 973, 982, 987, 1005, 1014, 1019

Offset: 1

Gus Wiseman, Oct 13 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ..., and the skew-alternating sum to be A - B - C + D + E - F - G + ...

For partitions and half only (or both): A000583, counted by A035363.
These compositions are counted by A228248.
For half-alternating only: A357625, reverse A357626, counted by A357641.
For skew-alternating only: A357627, reverse A357628, counted by A001700.
For reversed partitions and half only: A357631, counted by A357639.
For reversed partitions and skew only A357632, counted by A357640.
For partitions and skew only: A357636, counted by A035594.
Cf. A001511, A004006, A088218, A357136, A357182, A357642.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[0,1000],halfats[stc[#]]==0&&skats[stc[#]]==0&]

Intersection of A357625 and A357627.

cp's OEIS Frontend

A357624 Skew-alternating sum of the reversed n-th composition in standard order.

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A228248 Number of 2n-step lattice paths from (0,0) to (0,0) using steps in {N, S, E, W} starting with East, then always moving straight ahead or turning left.

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A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

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Mathematica

Formula