A099754 -id:A099754 - cp's OEIS frontend

A254028 a(n) = 2^(n+1) + 3^n + 3.

6, 10, 20, 46, 116, 310, 860, 2446, 7076, 20710, 61100, 181246, 539636, 1610710, 4815740, 14414446, 43177796, 129402310, 387944780, 1163310046, 3488881556, 10464547510, 31389448220, 94159956046, 282463090916, 847355718310

Offset: 0

Luciano Ancora, Jan 22 2015

This is the sequence of third terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A085279, A094374, A099754.

Table[2^(n+1)+3^n+3,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{6,10,20},30] (* Harvey P. Dale, Mar 27 2025 *)

a(n)=2<Charles R Greathouse IV, Jan 23 2015

Vec(-2*(13*x^2-13*x+3)/((x-1)*(2*x-1)*(3*x-1)) + O(x^100)) \\ Colin Barker, Jan 24 2015

G.f.: -2*(13*x^2-13*x+3) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Jan 24 2015
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - Colin Barker, Jan 24 2015
a(n) = A085279(n+1) = 2*( A099754(n)+1 ) = 2*( A094374(n)-2 ). [Bruno Berselli, Jan 26 2015]

A298119 Array read by antidiagonals: T(m,n) = number of Eulerian orientations of the torus grid graph C_m X C_n.

Original entry on oeis.org

4, 8, 8, 16, 18, 16, 32, 44, 44, 32, 64, 114, 148, 114, 64, 128, 308, 548, 548, 308, 128, 256, 858, 2116, 2970, 2116, 858, 256, 512, 2444, 8324, 16892, 16892, 8324, 2444, 512, 1024, 7074, 33028, 98466, 143224, 98466, 33028, 7074

Offset: 1

Andrew Howroyd, Jan 12 2018

In other words, the number of orientations of the m X n torus grid graph in which each vertex has equal indegree and outdegree.

Values are always even since reversing the orientation of each edge will always result in another Eulerian orientation.

			Array begins:
============================================================
m\n|   1    2     3      4        5         6          7
---|--------------------------------------------------------
1  |   4    8    16     32       64       128        256 ...
2  |   8   18    44    114      308       858       2444 ...
3  |  16   44   148    548     2116      8324      33028 ...
4  |  32  114   548   2970    16892     98466     583412 ...
5  |  64  308  2116  16892   143224   1250228   11091536 ...
6  | 128  858  8324  98466  1250228  16448400  220603364 ...
7  | 256 2444 33028 583412 11091536 220603364 4484823396 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..91
Eric Weisstein's World of Mathematics, Torus Grid Graph

Main diagonal is A054759.
Rows 2..5 are 2*A099754, 2*A170938, A298201, A372093, A372094.
Cf. A212801, A298117.

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Original entry on oeis.org

0, 2, 0, 3, 2, 7, 10, 23, 42, 87, 170, 343, 682, 1367, 2730, 5463, 10922, 21847, 43690, 87383, 174762, 349527, 699050, 1398103, 2796202, 5592407, 11184810, 22369623, 44739242, 89478487, 178956970, 357913943, 715827882, 1431655767, 2863311530, 5726623063, 11453246122, 22906492247, 45812984490

Offset: 0

Miklos Kristof, Dec 07 2007

Partial sums of A155980 for n > 2. - Klaus Purath, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A005578, A099754.
Cf. A001045, A327767, A328284.
Cf. A007583, A062092, A087289, A020988 (even bisection), A163834 (odd bisection), A078008, A084247, A181565.

List([0..40], n-> (2^n+3-7*(-1)^n+3*0^n)/6); # G. C. Greubel, Sep 02 2019

a135351:=func< n | (2^n+3-7*(-1)^n+3*0^n)/6 >; [ a135351(n): n in [0..32] ]; // Klaus Brockhaus, Dec 05 2009

G(x):=x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/n!,n=0..30);

Join[{0}, Table[(2^n +3 -7*(-1)^n)/6, {n,40}]] (* G. C. Greubel, Oct 11 2016 *)
LinearRecurrence[{2,1,-2},{0,2,0,3},40] (* Harvey P. Dale, Feb 13 2024 *)

a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; \\ Michel Marcus, Oct 11 2016

[(2^n+3-7*(-1)^n+3*0^n)/6 for n in (0..40)] # G. C. Greubel, Sep 02 2019

G.f.: x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)).
E.g.f.: (exp(2*x) + 3*exp(x) - 7*exp(-x) + 3)/6.
From Paul Curtz, Dec 20 2020: (Start)
a(n) + (period 2 sequence: repeat [1, -2]) = A328284(n+2).
a(n+1) + (period 2 sequence: repeat [-2, 1]) = A001045(n).
a(n+1) + (period 2 sequence: repeat [-1, 0]) = A078008(n).
a(n+1) + (period 2 sequence : repeat [-3, 2]) = -(-1)^n*A084247(n).
a(n+4) = a(n+1) + 7*A001045(n).
a(n+4) + a(n+1) = A181565(n).
a(2*n+2) + a(2*n+3) = A087289(n) = 3*A007583(n).
a(2*n+1) = A163834(n), a(2*n+2) = A020988(n). (End)

First part of definition corrected by Klaus Brockhaus, Dec 05 2009

cp's OEIS Frontend

A254028 a(n) = 2^(n+1) + 3^n + 3.

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A298119 Array read by antidiagonals: T(m,n) = number of Eulerian orientations of the torus grid graph C_m X C_n.

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A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

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cp's OEIS Frontend

A254028 a(n) = 2^(n+1) + 3^n + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A298119 Array read by antidiagonals: T(m,n) = number of Eulerian orientations of the torus grid graph C_m X C_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A135351 a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.