A094958 -id:A094958 - cp's OEIS frontend

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

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

A121451 Maximum product over partitions into parts of the form 3k+2.

Original entry on oeis.org

0, 2, 0, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576

Offset: 1

John W. Layman, Apr 26 2007

nonn

With the exception of the first three terms of this sequence and the first two terms of A094958, these two sequences appear to be identical.

			The only partition of 7 into parts of the form 3k+2 is {5,2}, so the maximum product is a(7)=10.

Cf. A000792, A034893, A094958.

Conjecture. a(1)=a(3)=0, otherwise a(n)=2^(n/2) if n is even and a(n)=5*2^((n-5)/2) if n is odd. (This has been verified for up to n=40.)

A128939 Maximal product over partitions of n into parts of the form 3k+1.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 7, 16, 16, 16, 28, 64, 64, 64, 112, 256, 256, 256, 448, 1024, 1024, 1024, 1792, 4096, 4096, 4096, 7168, 16384, 16384, 16384, 28672, 65536, 65536, 65536, 114688, 262144, 262144, 262144, 458752, 1048576

Offset: 1

John W. Layman, Apr 27 2007

nonn

Cf. A000792, A034893, A094958.

Conjecture. a(n)=1 if n<4, else a(n)=4^[n/4] if n<>4m+3, else a(n)=7*4^([n/4]-1), where [...] denotes the Floor function. (This has been verified up to n=40.)

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).

Original entry on oeis.org

3, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304, 5242880

Offset: 1

Arkadiusz Wesolowski, Aug 20 2013

Union of A020714 and A198633.
Essentially the same as A094958.
For every n, a(1)^3 + a(2)^3 + a(3)^3 + ... + a(2*n-1)^3 is a cube.

			a(9) = 40 because it is equal to 5*2^(4-1).

Wikipedia, Cube (algebra)
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A020714, A094958, A121451, A164682, A198633.

[n le 3 select n+2 else 2*Self(n-2) : n in [1..43]];

CoefficientList[Series[(3 + 4 x - x^2)/(1 - 2 x^2), {x, 0, 50}], x] (* Bruno Berselli, Aug 20 2013 *)

r=43; print1(3); print1(", "); for(n=2, r, if(bitand(n, 1), print1(5*2^((n-3)/2)), print1(2^(n/2+1))); print1(", "));

a(n) = ceiling((9 - (- 1)^n)*2^(floor(n/2) - 2)).
a(n) = n + 2 for n <= 3; a(n) = 2*a(n-2) for n > 3.
From Bruno Berselli, Aug 20 2013: (Start)
G.f.: x*(3+4*x-x^2)/(1-2*x^2).
a(n) = (16-(8-5*r)*(1-(-1)^n))*r^(n-6) for n>1, r=sqrt(2). (End)
E.g.f.: (8*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) + 2*x - 8)/4. - Stefano Spezia, Apr 09 2025

A356050 a(n) = 2*A135318(n+1) - A135318(n).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 11, 14, 21, 26, 43, 54, 85, 106, 171, 214, 341, 426, 683, 854, 1365, 1706, 2731, 3414, 5461, 6826, 10923, 13654, 21845, 27306, 43691, 54614, 87381, 109226, 174763, 218454, 349525, 436906, 699051, 873814, 1398101, 1747626, 2796203, 3495254, 5592405

Offset: 0

Paul Curtz, Aug 19 2022

Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Cf. A001045, A084214, A135318, A230096.

Cf. A094958.

LinearRecurrence[{0, 1, 0, 2}, {1, 1, 3, 4}, 50] (* Amiram Eldar, Aug 19 2022 *)

a(n) = A135318(n) + A230096(n+1).
a(n) = a(n-8) + 5*A094958(n-5).
a(2*n) = A001045(n+2).
a(2*n+1) = A084214(n+1).
From Stefano Spezia, Aug 20 2022: (Start)
O.g.f.: (1 + x + 2*x^2 + 3*x^3)/((1 + x^2)*(1 - 2*x^2)).
E.g.f.: (8*cosh(sqrt(2)*x) - 2*cos(x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 4*sin(x))/6. (End)
3*a(n) = A228826(n+1) +A094958(n+3). - R. J. Mathar, Jan 25 2023

cp's OEIS Frontend

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

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A121451 Maximum product over partitions into parts of the form 3k+2.

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A128939 Maximal product over partitions of n into parts of the form 3k+1.

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A228305 a(1) = 3; for n >= 1, a(2*n) = 2^(n+1), a(2*n+1) = 5*2^(n-1).

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A356050 a(n) = 2*A135318(n+1) - A135318(n).

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A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).