A029746 -id:A029746 - cp's OEIS frontend

A357621 Half-alternating sum of the n-th composition in standard order.

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

Offset: 0

Gus Wiseman, Oct 07 2022

sign

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 - ...

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 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 reverse version is A357622.
The skew-alternating form is A357623, reverse A357624.
Positions of zeros are A357625, reverse A357626.
The version for prime indices is A357629.
The version for Heinz numbers of partitions is A357633.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185.
Cf. A357627, A357628, A357630, A357631, A357634, A357635, A357640.

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]}];
Table[halfats[stc[n]],{n,0,100}]

Positions of first appearances are powers of 2 and even powers of 2 times 7, or A029746 without 7.

A029748 Numbers of the form 2^k times 1, 3 or 7.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 24, 28, 32, 48, 56, 64, 96, 112, 128, 192, 224, 256, 384, 448, 512, 768, 896, 1024, 1536, 1792, 2048, 3072, 3584, 4096, 6144, 7168, 8192, 12288, 14336, 16384, 24576, 28672, 32768, 49152, 57344, 65536, 98304, 114688, 131072

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Cf. A000079, A005009, A007283, A029744, A029746.

CoefficientList[Series[-(x^5 + 2 x^4 + 2 x^3 + 3 x^2 + 2 x + 1)/(2 x^3 - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Sort[Flatten[#{1,3,7}&/@(2^Range[0,20])]] (* Harvey P. Dale, Jan 30 2014 *)

From Colin Barker, Jul 19 2013: (Start)
a(n) = 2*a(n-3) for n>5.
G.f.: -(x^5+2*x^4+2*x^3+3*x^2+2*x+1) / (2*x^3-1). (End)
Sum_{n>=0} 1/a(n) = 62/21. - Amiram Eldar, Jan 17 2022

More terms from Colin Barker, Jul 19 2013

A029749 Numbers of the form 2^k times 1, 5 or 7.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 40, 56, 64, 80, 112, 128, 160, 224, 256, 320, 448, 512, 640, 896, 1024, 1280, 1792, 2048, 2560, 3584, 4096, 5120, 7168, 8192, 10240, 14336, 16384, 20480, 28672, 32768, 40960, 57344, 65536, 81920, 114688, 131072

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Cf. A000079, A005009, A029746, A029748, A094958.

CoefficientList[Series[-(3 x^4 + 3 x^3 + 4 x^2 + 2 x + 1) / (2 x^3 - 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)

From Colin Barker, Jul 19 2013: (Start)
a(n) = 2*a(n-3) for n>4.
G.f.: -(3*x^4 + 3*x^3 + 4*x^2 + 2*x + 1) / (2*x^3 - 1). (End)
Sum_{n>=0} 1/a(n) = 94/35. - Amiram Eldar, Jan 21 2022

More terms from Colin Barker, Jul 19 2013

cp's OEIS Frontend

A357621 Half-alternating sum of the n-th composition in standard order.

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A029748 Numbers of the form 2^k times 1, 3 or 7.

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A029749 Numbers of the form 2^k times 1, 5 or 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions