A246036 -id:A246036 - cp's OEIS frontend

A054881 Number of walks of length n along the edges of an octahedron starting and ending at a vertex and also ( with a(0)=0 ) between two opposite vertices.

1, 0, 4, 8, 48, 160, 704, 2688, 11008, 43520, 175104, 698368, 2797568, 11182080, 44744704, 178946048, 715849728, 2863267840, 11453333504, 45812809728, 183252287488, 733007052800, 2932032405504, 11728121233408

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 7.
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003683, A054882, A054883, A054884, A054885, A246036.

[(4^n+(-1)^n*2^(n+1)+3*0^n)/6: n in [0..30]]; // Vincenzo Librandi, Apr 23 2015

CoefficientList[Series[(1-2*x-4*x^2)/((1+2x)*(1-4x)), {x,0,40}], x] (* L. Edson Jeffery, Apr 22 2015 *)
LinearRecurrence[{2,8}, {1,0,4}, 41] (* G. C. Greubel, Feb 06 2023 *)

[(4^n + (-1)^n*2^(n+1) + 3*0^n)/6 for n in range(31)] # G. C. Greubel, Feb 06 2023

a(n) = 4*A003683(n-1) + 0^n/2, n >= 0.
a(n) = (4^n + (-1)^n*2^(n+1) + 3*0^n)/6.
G.f.: (1/6)*(3 + 2/(1+2*x) + 1/(1-4*x)).
From L. Edson Jeffery, Apr 22 2015: (Start)
G.f.: (1-2*x-4*x^2)/((1+2*x)*(1-4*x)).
a(n) = 8*A246036(n-3) + 0^n/2, n >= 0. (End)
a(n) = 2^n*A001045(n-1) + (1/2)*[n=0] = 2^n*(2^(n-1) + (-1)^n)/3 + (1/2)*[n=0], n >= 0. - Ralf Steiner, Aug 27 2020, edited by M. F. Hasler, Sep 11 2020
E.g.f.: (1/6)*(exp(4*x) + 2*exp(-2*x) + 3). - G. C. Greubel, Feb 06 2023

A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0

Offset: 0

Paul Curtz, Jul 24 2008

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

			Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

[2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023

A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009

T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022

def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Edited and extended by R. J. Mathar, Sep 07 2009

A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

Original entry on oeis.org

1, 6, 6, 20, 6, 36, 20, 88, 6, 36, 36, 120, 20, 120, 88, 336, 6, 36, 36, 120, 36, 216, 120, 528, 20, 120, 120, 400, 88, 528, 336, 1376, 6, 36, 36, 120, 36, 216, 120, 528, 36, 216, 216, 720, 120, 720, 528, 2016, 20, 120, 120, 400, 120, 720, 400, 1760, 88, 528, 528, 1760, 336, 2016, 1376, 5440

Offset: 0

N. J. A. Sloane, Aug 21 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
This is the odd-rule cellular automaton defined by OddRule 077 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of A246036.
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

			Here is the neighborhood:
[X, X, X]
[0, 0, 0]
[X, X, X]
which contains a(1) = 6 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8192
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.

Cf. A246036.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=(1/x+1+x)*(1/y+y);
OddCA(f, 70);

(* f = A246036 *) f[0] = 1; f[n_] := (4^(n+1)-(-2)^n)/3; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

A271494 Expansion of (1+16x)/((1+4x)(1-8x)).

Original entry on oeis.org

1, 20, 112, 1088, 7936, 66560, 520192, 4210688, 33488896, 268697600, 2146435072, 17184063488, 137422176256, 1099578736640, 8795824586752, 70369817919488, 562945658454016, 4503616807239680, 36028728299487232, 288230651029618688, 2305841909702066176, 18446748471756062720

Offset: 0

N. J. A. Sloane, Apr 15 2016

Sixth moments of the Rudin-Shapiro polynomials.

Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
Doron Zeilberger, Personal Communication to N. J. A. Sloane, Apr 15 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.
Index entries for linear recurrences with constant coefficients, signature (4,32).

Cf. A246036, A271495, A271496.

CoefficientList[Series[(1+16x)/((1+4x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,32},{1,20},30] (* Harvey P. Dale, May 13 2017 *)

Vec((1+16*x)/((1+4*x)*(1-8*x)) + O(x^50)) \\ Colin Barker, Apr 17 2016

From Colin Barker, Apr 17 2016: (Start)
a(n) = 2^(1+3*n)-(-4)^n.
a(n) = 4*a(n-1) + 32*a(n-2) for n>1.
(End)
a(n) = 4^n*A014551(n+1). - R. J. Mathar, Mar 08 2021

A271495 Eighth moments of the Rudin-Shapiro polynomials.

Original entry on oeis.org

1, 70, 668, 14104, 198640, 3420256, 53143488, 864838016, 13714054912, 220102985216, 3513567575040, 56284226394112, 900460612808704, 14414430456537088, 230619566685274112, 3689872453256970240, 59031392914560188416, 944463240632040030208, 15111217402853747064832

Offset: 0

N. J. A. Sloane, Apr 15 2016

nonn

Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
Doron Zeilberger, Personal Communication to N. J. A. Sloane, Apr 15 2016.

Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.

Cf. A246036, A271494, A271496.

G.f.: -(90194313216*t^11 -15300820992*t^10 -1979711488*t^9 -292552704*t^8 -22216704*t^7 +10649600*t^6 -1024*t^5 -144384*t^4 +7008*t^3 +664*t^2 -54*t -1) / (8*t+1) / (16*t-1) / (1409286144*t^10 -264241152*t^9 -25690112*t^8 -4128768*t^7 -311296*t^6 +170496*t^5 -2624*t^4 -2208*t^3 +148*t^2 +8*t -1).

A271496 Tenth moments of the Rudin-Shapiro polynomials.

Original entry on oeis.org

1, 252, 4144, 187328, 5143296, 182336512, 5664518144, 185202884608, 5854499373056, 187705543360512, 5987197055401984, 191926028833128448, 6145480467693961216, 196854507263106220032, 6299857999165520871424, 201565743393381566906368, 6448321065980862740299776

Offset: 0

N. J. A. Sloane, Apr 15 2016

nonn

Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
Doron Zeilberger, Personal Communication to N. J. A. Sloane, Apr 15 2016.

Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.

Cf. A246036, A271494, A271495.

G.f.: -(369435906932736*t^11 -32160715112448*t^10 -2001454759936*t^9 -145223581696*t^8 -4454350848*t^7 +1392508928*t^6 -5865472*t^5 -4599808*t^4 +123648*t^3 +4768*t^2 -220*t -1) / (1+16*t) / (32*t-1) / (1443109011456*t^10 -135291469824*t^9 -6576668672*t^8 -528482304*t^7 -19922944*t^6 +5455872*t^5 -41984*t^4 -17664*t^3 +592*t^2 +16*t -1)

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249

Offset: 0

Paul Curtz, Mar 29 2022

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
    4,  -3,   5,  -1,   9,   7, 25, ...
   -7,   8,  -6,  10,  -2,  18, 14,  50, ...
   15, -14,  16, -12,  20,  -4, 36,  28, 100, ...
  -29,  30, -28,  32, -24,  40, -8,  72,  56, 200, ...
   59, -58,  60, -56,  64, -48, 80, -16, 144, 112, 400, ...
  ...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n)   - a(n) =  0,  0,  0,  0,   0, ...       (trivially)
a(n+1) + a(n) =  1,  2,  4,  8,  16, ... = 2^n (by definition)
a(n+2) - a(n) =  1,  2,  4,  8,  16, ... = 2^n
a(n+3) + a(n) =  3,  6, 12, 24,  48, ... = 2^n*3
a(n+4) - a(n) =  5, 10, 20, 40,  80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
     (...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
   4,   3,   5,    1,    9,   -7,   25,   -39, ...  = c(n)
  -1,   2,  -4,    8,  -16,   32,  -64,   128, ...  = -A122803(n)
   3,  -6,  12,  -24,   48,  -96,  192,  -384, ...  =
  -9,  18, -36,   72, -144,  288, -576,  1152, ...
  27, -54, 108, -216,  432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

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

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022

LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)

a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022

Edited by M. F. Hasler, Apr 26 2022.

cp's OEIS Frontend

A054881 Number of walks of length n along the edges of an octahedron starting and ending at a vertex and also ( with a(0)=0 ) between two opposite vertices.

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A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

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A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

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A271494 Expansion of (1+16*x)/((1+4*x)*(1-8*x)).

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A271495 Eighth moments of the Rudin-Shapiro polynomials.

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A271496 Tenth moments of the Rudin-Shapiro polynomials.

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A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

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A271494 Expansion of (1+16x)/((1+4x)(1-8x)).