A066099 -id:A066099 - cp's OEIS frontend

A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

9, 34, 39, 45, 49, 57, 132, 139, 142, 149, 154, 159, 161, 169, 178, 183, 189, 194, 199, 205, 209, 217, 226, 231, 237, 241, 249, 520, 531, 534, 540, 549, 554, 559, 564, 571, 574, 577, 585, 594, 599, 605, 612, 619, 622, 629, 634, 639, 642, 647, 653, 657, 665

Offset: 1

Gus Wiseman, Jul 10 2021

nonn

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

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 initial terms and the corresponding compositions:
      9: (3,1)            183: (2,1,2,1,1,1)
     34: (4,2)            189: (2,1,1,1,2,1)
     39: (3,1,1,1)        194: (1,5,2)
     45: (2,1,2,1)        199: (1,4,1,1,1)
     49: (1,4,1)          205: (1,3,1,2,1)
     57: (1,1,3,1)        209: (1,2,4,1)
    132: (5,3)            217: (1,2,1,3,1)
    139: (4,2,1,1)        226: (1,1,4,2)
    142: (4,1,1,2)        231: (1,1,3,1,1,1)
    149: (3,2,2,1)        237: (1,1,2,1,2,1)
    154: (3,1,2,2)        241: (1,1,1,4,1)
    159: (3,1,1,1,1,1)    249: (1,1,1,1,3,1)
    161: (2,5,1)          520: (6,4)
    169: (2,2,3,1)        531: (5,3,1,1)
    178: (2,1,3,2)        534: (5,2,1,2)

These compositions are counted by A088218.
These are the positions of 2's in A344618.
The case of partitions of 2n is A344741.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, A344743.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]==-2&]

A345924 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.

Original entry on oeis.org

12, 40, 49, 51, 54, 60, 144, 161, 163, 166, 172, 184, 194, 197, 199, 202, 205, 207, 212, 217, 219, 222, 232, 241, 243, 246, 252, 544, 577, 579, 582, 588, 600, 624, 642, 645, 647, 650, 653, 655, 660, 665, 667, 670, 680, 689, 691, 694, 700, 720, 737, 739, 742

Offset: 1

Gus Wiseman, Jul 11 2021

nonn

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

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 initial terms and the corresponding compositions:
     12: (1,3)          202: (1,3,2,2)        582: (3,4,1,2)
     40: (2,4)          205: (1,3,1,2,1)      588: (3,3,1,3)
     49: (1,4,1)        207: (1,3,1,1,1,1)    600: (3,2,1,4)
     51: (1,3,1,1)      212: (1,2,2,3)        624: (3,1,1,5)
     54: (1,2,1,2)      217: (1,2,1,3,1)      642: (2,6,2)
     60: (1,1,1,3)      219: (1,2,1,2,1,1)    645: (2,5,2,1)
    144: (3,5)          222: (1,2,1,1,1,2)    647: (2,5,1,1,1)
    161: (2,5,1)        232: (1,1,2,4)        650: (2,4,2,2)
    163: (2,4,1,1)      241: (1,1,1,4,1)      653: (2,4,1,2,1)
    166: (2,3,1,2)      243: (1,1,1,3,1,1)    655: (2,4,1,1,1,1)
    172: (2,2,1,3)      246: (1,1,1,2,1,2)    660: (2,3,2,3)
    184: (2,1,1,4)      252: (1,1,1,1,1,3)    665: (2,3,1,3,1)
    194: (1,5,2)        544: (4,6)            667: (2,3,1,2,1,1)
    197: (1,4,2,1)      577: (3,6,1)          670: (2,3,1,1,1,2)
    199: (1,4,1,1,1)    579: (3,5,1,1)        680: (2,2,2,4)

These compositions are counted by A002054.
These are the positions of -2's in A124754.
The version for reverse-alternating sum is A345923.
The opposite (positive 2) version is A345925.
The version for Heinz numbers of partitions is A345962.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A008549, A025047, A163493, A239830, A344609, A344651, A344741, A345908, A345961.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]==-2&]

A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.

Original entry on oeis.org

2, 9, 11, 14, 34, 37, 39, 42, 45, 47, 52, 57, 59, 62, 132, 137, 139, 142, 146, 149, 151, 154, 157, 159, 164, 169, 171, 174, 178, 181, 183, 186, 189, 191, 200, 209, 211, 214, 220, 226, 229, 231, 234, 237, 239, 244, 249, 251, 254, 520, 529, 531, 534, 540, 546

Offset: 1

Gus Wiseman, Jul 11 2021

nonn

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

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 initial terms and corresponding compositions:
      2: (2)            137: (4,3,1)
      9: (3,1)          139: (4,2,1,1)
     11: (2,1,1)        142: (4,1,1,2)
     14: (1,1,2)        146: (3,3,2)
     34: (4,2)          149: (3,2,2,1)
     37: (3,2,1)        151: (3,2,1,1,1)
     39: (3,1,1,1)      154: (3,1,2,2)
     42: (2,2,2)        157: (3,1,1,2,1)
     45: (2,1,2,1)      159: (3,1,1,1,1,1)
     47: (2,1,1,1,1)    164: (2,3,3)
     52: (1,2,3)        169: (2,2,3,1)
     57: (1,1,3,1)      171: (2,2,2,1,1)
     59: (1,1,2,1,1)    174: (2,2,1,1,2)
     62: (1,1,1,1,2)    178: (2,1,3,2)
    132: (5,3)          181: (2,1,2,2,1)

These compositions are counted by A088218.
These are the positions of 2's in A124754.
The case of partitions of 2n is A344741.
The version for reverse-alternating sum is A345922.
The opposite (negative 2) version is A345924.
The version for Heinz numbers of partitions is A345960 (reverse: A345961).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A025047, A114121, A163493, A238279, A239830, A344607, A344608, A344609, A344651, A344743.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]==2&]

A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 12, 14, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 37, 38, 40, 42, 44, 47, 48, 51, 52, 54, 56, 59, 60, 62, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88, 91, 92, 93, 94, 96, 99, 100, 101, 102, 104, 106, 107, 108

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

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

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 initial terms and the corresponding compositions:
     1: (1)        26: (1,2,2)        52: (1,2,3)
     2: (2)        27: (1,2,1,1)      54: (1,2,1,2)
     4: (3)        28: (1,1,3)        56: (1,1,4)
     6: (1,2)      30: (1,1,1,2)      59: (1,1,2,1,1)
     7: (1,1,1)    31: (1,1,1,1,1)    60: (1,1,1,3)
     8: (4)        32: (6)            62: (1,1,1,1,2)
    11: (2,1,1)    35: (4,1,1)        64: (7)
    12: (1,3)      37: (3,2,1)        67: (5,1,1)
    14: (1,1,2)    38: (3,1,2)        69: (4,2,1)
    16: (5)        40: (2,4)          70: (4,1,2)
    19: (3,1,1)    42: (2,2,2)        72: (3,4)
    20: (2,3)      44: (2,1,3)        73: (3,3,1)
    21: (2,2,1)    47: (2,1,1,1,1)    74: (3,2,2)
    22: (2,1,2)    48: (1,5)          76: (3,1,3)
    24: (1,4)      51: (1,3,1,1)      79: (3,1,1,1,1)

The version for prime indices is A000037.
The version for Heinz numbers of partitions is A026424, counted by A027193.
These compositions are counted by A027306.
These are the positions of terms > 0 in A344618.
The weak (k >= 0) version is A345914.
The version for unreversed alternating sum is A345917.
The opposite (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000290, A000346, A008549, A025047, A027187, A028260, A034871, A114121, A163493, A344607, A344608, A344650, A344743, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]>0&]

A345920 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.

Original entry on oeis.org

5, 9, 17, 18, 23, 25, 29, 33, 34, 39, 45, 49, 57, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 139, 141, 142, 149, 153, 154, 159, 161, 169, 177, 178, 183, 189, 193, 194, 199, 205, 209, 217

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

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

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 initial terms and the corresponding compositions:
      5: (2,1)         68: (4,3)
      9: (3,1)         71: (4,1,1,1)
     17: (4,1)         75: (3,2,1,1)
     18: (3,2)         77: (3,1,2,1)
     23: (2,1,1,1)     78: (3,1,1,2)
     25: (1,3,1)       81: (2,4,1)
     29: (1,1,2,1)     85: (2,2,2,1)
     33: (5,1)         89: (2,1,3,1)
     34: (4,2)         90: (2,1,2,2)
     39: (3,1,1,1)     95: (2,1,1,1,1,1)
     45: (2,1,2,1)     97: (1,5,1)
     49: (1,4,1)       98: (1,4,2)
     57: (1,1,3,1)    103: (1,3,1,1,1)
     65: (6,1)        105: (1,2,3,1)
     66: (5,2)        109: (1,2,1,2,1)

The version for prime indices is {}.
The version for Heinz numbers of partitions is A119899.
These compositions are counted by A294175  (even bisection: A008549).
These are the positions of terms < 0 in A344618.
The complement is A345914.
The weak (k <= 0) version is A345916.
The opposite (k > 0) version is A345918.
The version for unreversed alternating sum is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]<0&]

A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jul 10 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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.
Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.

			The initial terms and the corresponding compositions:
     1: (1)        20: (2,3)          35: (4,1,1)
     2: (2)        21: (2,2,1)        37: (3,2,1)
     4: (3)        22: (2,1,2)        38: (3,1,2)
     5: (2,1)      23: (2,1,1,1)      39: (3,1,1,1)
     6: (1,2)      24: (1,4)          40: (2,4)
     7: (1,1,1)    25: (1,3,1)        42: (2,2,2)
     8: (4)        26: (1,2,2)        44: (2,1,3)
     9: (3,1)      27: (1,2,1,1)      45: (2,1,2,1)
    11: (2,1,1)    28: (1,1,3)        47: (2,1,1,1,1)
    12: (1,3)      29: (1,1,2,1)      48: (1,5)
    14: (1,1,2)    30: (1,1,1,2)      49: (1,4,1)
    16: (5)        31: (1,1,1,1,1)    51: (1,3,1,1)
    17: (4,1)      32: (6)            52: (1,2,3)
    18: (3,2)      33: (5,1)          54: (1,2,1,2)
    19: (3,1,1)    34: (4,2)          56: (1,1,4)

The version for Heinz numbers of partitions is A000037.
These compositions are counted by A058622.
These are the positions of terms != 0 in A124754.
The complement (k = 0) is A344619.
The positive (k > 0) version is A345917 (reverse: A345918).
The negative (k < 0) version is A345919 (reverse: A345920).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]!=0&]

A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88

Offset: 1

Gus Wiseman, Jul 04 2021

nonn

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

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 sequence of terms together with the corresponding compositions begins:
     0: ()           19: (3,1,1)        40: (2,4)
     1: (1)          20: (2,3)          41: (2,3,1)
     2: (2)          21: (2,2,1)        42: (2,2,2)
     3: (1,1)        22: (2,1,2)        43: (2,2,1,1)
     4: (3)          24: (1,4)          44: (2,1,3)
     6: (1,2)        26: (1,2,2)        46: (2,1,1,2)
     7: (1,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
     8: (4)          28: (1,1,3)        48: (1,5)
    10: (2,2)        30: (1,1,1,2)      50: (1,3,2)
    11: (2,1,1)      31: (1,1,1,1,1)    51: (1,3,1,1)
    12: (1,3)        32: (6)            52: (1,2,3)
    13: (1,2,1)      35: (4,1,1)        53: (1,2,2,1)
    14: (1,1,2)      36: (3,3)          54: (1,2,1,2)
    15: (1,1,1,1)    37: (3,2,1)        55: (1,2,1,1,1)
    16: (5)          38: (3,1,2)        56: (1,1,4)

The version for prime indices is A000027, counted by A000041.
These compositions are counted by A116406.
The case of non-Heinz numbers of partitions is A119899, counted by A344608.
The version for Heinz numbers of partitions is A344609, counted by A344607.
These are the positions of terms >= 0 in A344618.
The version for unreversed alternating sum is A345913.
The opposite (k <= 0) version is A345916.
The strict (k > 0) case is A345918.
The complement is A345920, counted by A294175.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]>=0&]

A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

Original entry on oeis.org

0, 3, 6, 10, 12, 13, 15, 20, 24, 25, 27, 30, 36, 40, 41, 43, 46, 48, 49, 50, 51, 53, 54, 55, 58, 60, 61, 63, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 136, 144, 145, 147, 150, 156, 160, 161, 162, 163

Offset: 1

Gus Wiseman, Jul 08 2021

nonn

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

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 sequence of terms together with the corresponding compositions begins:
     0: ()
     3: (1,1)
     6: (1,2)
    10: (2,2)
    12: (1,3)
    13: (1,2,1)
    15: (1,1,1,1)
    20: (2,3)
    24: (1,4)
    25: (1,3,1)
    27: (1,2,1,1)
    30: (1,1,1,2)
    36: (3,3)
    40: (2,4)
    41: (2,3,1)

The version for Heinz numbers of partitions is A028260 (counted by A027187).
These compositions are counted by A058622.
These are the positions of terms <= 0 in A124754.
The reverse-alternating version is A345916.
The opposite (k >= 0) version is A345917.
The strictly negative (k < 0) version is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A008549, A025047, A032443, A114121, A163493, A344607, A344609, A344610.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]<=0&]

A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.

Original entry on oeis.org

0, 3, 5, 9, 10, 13, 15, 17, 18, 23, 25, 29, 33, 34, 36, 39, 41, 43, 45, 46, 49, 50, 53, 55, 57, 58, 61, 63, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 136, 139, 141, 142, 145, 147, 149

Offset: 1

Gus Wiseman, Jul 08 2021

nonn

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

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 sequence of terms together with the corresponding compositions begins:
     0: ()
     3: (1,1)
     5: (2,1)
     9: (3,1)
    10: (2,2)
    13: (1,2,1)
    15: (1,1,1,1)
    17: (4,1)
    18: (3,2)
    23: (2,1,1,1)
    25: (1,3,1)
    29: (1,1,2,1)
    33: (5,1)
    34: (4,2)
    36: (3,3)

The version for Heinz numbers of partitions is A000290.
These compositions are counted by A058622.
These are the positions of terms <= 0 in A344618.
The opposite (k >= 0) version is A345914.
The version for unreversed alternating sum is A345915.
The strictly negative (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A027187, A028260, A032443, A114121, A163493, A344607, A344610, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]<=0&]

A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97

Offset: 1

Gus Wiseman, May 28 2020

nonn

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 sequence together with the corresponding compositions begins:
   1: (1)          20: (2,3)          48: (1,5)
   2: (2)          23: (2,1,1,1)      49: (1,4,1)
   3: (1,1)        24: (1,4)          50: (1,3,2)
   4: (3)          25: (1,3,1)        51: (1,3,1,1)
   5: (2,1)        27: (1,2,1,1)      52: (1,2,3)
   6: (1,2)        28: (1,1,3)        55: (1,2,1,1,1)
   7: (1,1,1)      29: (1,1,2,1)      56: (1,1,4)
   8: (4)          30: (1,1,1,2)      57: (1,1,3,1)
   9: (3,1)        31: (1,1,1,1,1)    59: (1,1,2,1,1)
  11: (2,1,1)      32: (6)            60: (1,1,1,3)
  12: (1,3)        33: (5,1)          61: (1,1,1,2,1)
  13: (1,2,1)      35: (4,1,1)        62: (1,1,1,1,2)
  14: (1,1,2)      37: (3,2,1)        63: (1,1,1,1,1,1)
  15: (1,1,1,1)    38: (3,1,2)        64: (7)
  16: (5)          39: (3,1,1,1)      65: (6,1)
  17: (4,1)        41: (2,3,1)        66: (5,2)
  18: (3,2)        44: (2,1,3)        67: (5,1,1)
  19: (3,1,1)      47: (2,1,1,1,1)    68: (4,3)

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

The version counting partitions is A051424, with strict case A007360.
The version for binary indices is A087087.
The version counting compositions is A101268.
The version for prime indices is A302569.
The case without singletons is A333227.
The complement is A335236.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
All of the following pertain to compositions in standard order:
- Length is A000120.
- The parts are row k of A066099.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A048793, A272919, A291166, A302569, A335236, A335237, A335238, A335239, A335240.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==1||CoprimeQ@@stc[#]&]

cp's OEIS Frontend

A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

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A345924 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.

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A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.

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A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.

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A345920 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.

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A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

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A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

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A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

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A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.

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