A357182 -id:A357182 - cp's OEIS frontend

A222763 Number of n X 2 0..1 arrays with exactly floor(nX2/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

1, 0, 3, 4, 20, 48, 175, 512, 1719, 5400, 17776, 57420, 188656, 617176, 2033175, 6697744, 22139780, 73262232, 242931321, 806516560, 2681475048, 8925158440, 29740390672, 99196158144, 331163178475, 1106489052968, 3699881730900, 12380449027324, 41454579098852

Offset: 0

R. H. Hardin, Mar 05 2013

nonn

From Gus Wiseman, Oct 05 2022: (Start)
Conjecture: Also the number of integer compositions of 2n + 1 with the same length as reverse-alternating sum. Here, the reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i. For example, the a(4) = 20 compositions are:
  (135)    (234)    (333)    (432)    (531)
  (11115)  (21114)  (31113)  (41112)  (51111)
  (11214)  (21213)  (31212)  (41211)
  (11313)  (21312)  (31311)
  (11412)  (21411)
  (11511)
This is the odd-indexed version of A357182, and the corresponding unordered count (partitions) is A357488.
(End)

			All solutions for n=3
..0..1....0..0....0..0....0..0
..0..0....0..0....0..1....1..0
..0..0....1..0....0..0....0..0

Alois P. Heinz, Table of n, a(n) for n = 0..1881 (terms n = 1..210 from R. H. Hardin)

Column k = 2 of A222769.

Cf. A046736, A105422.

a:= proc(n) option remember; `if`(n<3, [1, 0, 3][n+1],
      (4*(n-1)*(74*n^2-153*n+73)*a(n-1) +8*(2*n-3)*
      (74*n^2-153*n+70)*a(n-2) -2*(37*n-21)*(2*n-5)*
      (n-1)*a(n-3))/(5*(37*n-58)*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 24 2024

a(n) = A105422(2n,n). - Alois P. Heinz, Sep 24 2024

a(n) = (n+1)*A046736(n+2). - Mark van Hoeij, Nov 29 2024

a(0)=1 prepended by Alois P. Heinz, Sep 24 2024

A357847 Number of integer compositions of n whose length is twice their alternating sum.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 3, 1, 8, 11, 15, 46, 59, 127, 259, 407, 888, 1591, 2925, 5896, 10607, 20582, 39446, 73448, 142691, 269777, 513721, 988638, 1876107, 3600313, 6893509, 13165219, 25288200, 48408011, 92824505, 178248758, 341801149, 656641084, 1261298356

Offset: 0

Gus Wiseman, Oct 16 2022

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(0) = 1 through a(9) = 15 compositions:
  ()  .  .  (21)  .  (32)  (1131)  (43)  (1142)  (54)
                           (2121)        (1241)  (111141)
                           (3111)        (2132)  (112131)
                                         (2231)  (113121)
                                         (3122)  (114111)
                                         (3221)  (211131)
                                         (4112)  (212121)
                                         (4211)  (213111)
                                                 (311121)
                                                 (312111)
                                                 (411111)

Alois P. Heinz, Table of n, a(n) for n = 0..3465

The version for partitions is A357709, ranked by A357848.
A011782 counts compositions.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
A357189 counts partitions w/ length = alternating sum, ranked by A357486.
Cf. A262977, A301987, A357183, A357485, A357488.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,10}]

a(21)-a(38) from Alois P. Heinz, Oct 19 2022

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 + ...

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.

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.

A357709 Number of integer partitions of n whose length is twice their alternating sum.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 6, 6, 9, 11, 13, 18, 21, 28, 32, 44, 49, 65, 76, 96, 114, 141, 170, 204, 250, 295, 361, 425, 516, 606, 734, 858, 1031, 1210, 1440, 1690, 2000, 2347, 2759, 3240, 3786, 4441, 5174, 6053, 7030, 8210, 9509, 11074, 12807, 14870

Offset: 0

Gus Wiseman, Oct 16 2022

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The alternating sum of a partition is also the number of odd conjugate parts.

			The a(1) = 0 through a(12) = 6 partitions:
  .  .  21  .  32  3111  43  3221  54      3331  65      4332
                             4211  411111  4222  422111  4431
                                           4321  521111  5322
                                           5311          5421
                                                         6411
                                                         51111111

This is the "twice" version of A357189, ranked by A357486.
The version for compositions is A357847.
These partitions are ranked by A357848.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
Cf. A004526, A262046, A262977, A301987, A357183, A357485, A357488.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,30}]

A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

Original entry on oeis.org

1, 6, 15, 35, 40, 77, 84, 90, 143, 189, 210, 220, 221, 224, 250, 323, 364, 437, 462, 490, 495, 504, 525, 528, 667, 748, 819, 858, 899, 988, 1029, 1040, 1134, 1147, 1155, 1188, 1210, 1320, 1326, 1375, 1400, 1408, 1517, 1564, 1683, 1690, 1763, 1904, 1938, 2021

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    15: {2,3}
    35: {3,4}
    40: {1,1,1,3}
    77: {4,5}
    84: {1,1,2,4}
    90: {1,2,2,3}
   143: {5,6}
   189: {2,2,2,4}
   210: {1,2,3,4}
   220: {1,1,3,5}
   221: {6,7}
   224: {1,1,1,1,1,4}

These partitions are counted by A357709.
The version for compositions is counted by A357847.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A025047 counts alternating compositions.
A056239 adds up prime indices.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
A357189 counts partitions w/ length = alternating sum, ranked by A357486.
Cf. A000720, A001221, A001222, A262977, A301987, A357183, A357485, A357488.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[1000],Length[primeMS[#]]==2sats[primeMS[#]]&]

cp's OEIS Frontend

A222763 Number of n X 2 0..1 arrays with exactly floor(nX2/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A357847 Number of integer compositions of n whose length is twice their alternating sum.

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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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A357709 Number of integer partitions of n whose length is twice their alternating sum.

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A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

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