A003683 -id:A003683 - cp's OEIS frontend

A084175 Jacobsthal oblong numbers.

0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215, 62549992960455

Offset: 0

Paul Barry, May 18 2003

Inverse binomial transform is A001019 doubled up.
Binomial transform is A084177.
Partial sums of A003683.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 11.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Except for initial terms, same as A015249 and A084152.

Cf. A001654, A001656, A084158, A084159, A099930.

List([0..30], n-> (2^(2*n+1) -(-2)^n -1)/9); # G. C. Greubel, Sep 21 2019

[(2*4^n-(-2)^n-1)/9: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

for n from 1 to 25 do print(round(2^n/3)*round(2^(n+1)/3)) od; # Gary Detlefs, Feb 10 2010

Table[(2*4^n -(-2)^n -1)/9, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011, modified by G. C. Greubel, Sep 21 2019 *)
LinearRecurrence[{3,6,-8}, {0,1,3}, 25] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=(2*4^n-(-2)^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

def A084175(n): return ((m:=1<Chai Wah Wu, Apr 25 2025

[gaussian_binomial(n, 2, -2) for n in range(1, 26)] # Zerinvary Lajos, May 28 2009

a(n) = A001045(n)*A001045(n+1).
a(n) = (2*4^n - (-2)^n - 1)/9;
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), a(0)=0, a(1)=1, a(2)=3.
G.f.: x/((1+2*x)*(1-x)*(1-4*x)).
E.g.f.: (2*exp(4*x) - exp(x) - exp(-2*x))/9.
a(n+1) - 4*a(n) = 1, -1, 3, -5, 11, ... = A001045(n+1) signed. - Paul Curtz, May 19 2008
a(n) = round(2^n/3) * round(2^(n+1)/3). - Gary Detlefs, Feb 10 2010
From Peter Bala, Mar 30 2015: (Start)
The shifted o.g.f. A(x) := 1/( (1 + 2*x)*(1 - x)*(1 - 4*x) ) = 1/(1 - 3*x - 6*x^2 + 8*x^3). Hence A(x) == 1/(1 - 3*x + 3*x^2 - x^3) (mod 9) == 1/(1 - x)^3 (mod 9). It follows by Theorem 1 of Heninger et al. that (A(x))^(1/3) = 1 + x + 4*x^2 + 10*x^3 + ... has integral coefficients.
Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} J(3*n)/J(n)*x^n/n ), where J(n) = A001045(n) are the Jacobsthal numbers. Cf. A001656, A099930. (End)

A091914 a(n) = 2a(n-1) + 12a(n-2).

Original entry on oeis.org

1, 2, 16, 56, 304, 1280, 6208, 27776, 130048, 593408, 2747392, 12615680, 58200064, 267788288, 1233977344, 5681414144, 26170556416, 120518082560, 555082842112, 2556382674944, 11773759455232, 54224111009792, 249733335482368

Offset: 0

Paul Barry, Feb 12 2004

Binomial transform of 1, 1, 13, 13, 169, 169, ....

The inverse binomial transform of 2^n*c(n), where c(n) is the solution to c(n) = c(n-1) + k*c(n-2), a(0)=1, a(1)=1 is 1, 1, 4k+1, 4k+1, (4k+1)^2, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,12).

Cf. A003683, A063727.

a := [1,2];; for n in [3..30] do a[n] := 2*a[n-1] + 12*a[n-2]; od; a; # Muniru A Asiru, Jan 31 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/(1-2*x-12*x^2))) // G. C. Greubel, Jan 30 2018

a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then 2*procname(n-1) + 12*procname(n-2) fi; end: # Muniru A Asiru, Jan 31 2018

LinearRecurrence[{2,12},{1,2},30] (* or *) With[{s=Sqrt[13]},Table[ Simplify[ -(((13+s)((1-s)^n-(1+s)^n))/(26(1+s)))],{n,30}]] (* Harvey P. Dale, May 25 2013 *)

my(x='x+O('x^30)); Vec(1/(1-2*x-12*x^2)) \\ G. C. Greubel, Jan 30 2018

[lucas_number1(n,2,-12) for n in range(1, 30)] # Zerinvary Lajos, Apr 22 2009

a(n) = A000079(n)*A006130(n).
G.f.: 1/(1-2*x-12*x^2).
a(n) = ((1+sqrt(13))*(1+sqrt(13))^n - (1-sqrt(13))*(1-sqrt(13))^n) /(2*sqrt(13)).
a(n) = Sum_{k=0..floor(n/2)} C(n+1,2*k+1) * 13^k. - Paul Barry, Jan 15 2007

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Original entry on oeis.org

1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360

Offset: 1

Clark Kimberling, Feb 18 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - Zagros Lalo, Jul 31 2018

			First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
    1
    2,   0
    4,   1,  0
    8,   4,  0, 0
   16,  12,  1, 0, 0
   32,  32,  6, 0, 0, 0
   64,  80, 24, 1, 0, 0, 0
  128, 192, 80, 8, 0, 0, 0, 0

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
S. Halici, On some Pell polynomials , Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

Cf. A028297, A207537, A133156, A038207, A099089.
Cf. A053117, A097610, A091894.
Cf. A013609, A038207.
Cf. A128099.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* Zagros Lalo, Jul 31 2018 *)

u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
From Philippe Deléham, Mar 04 2012: (Start)
With 0<=k<=n:
Mirror image of triangle in A099089.
Skew version of A038207.
Riordan array (1/(1-2*x), x^2/(1-2*x)).
G.f.: 1/(1-2*x-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
T(n,k) = A013609(n-k, n-2*k+1). - Johannes W. Meijer, Sep 05 2013
From Tom Copeland, Feb 11 2016: (Start)
A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
(End)

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.

Original entry on oeis.org

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

A143683 Pascal-(1,8,1) array.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 10,   19,    28,     37,      46,       55, ... A017173;
  1, 19,  118,   298,    559,     901,     1324, ...
  1, 28,  298,  1540,   4483,    9856,    18388, ...
  1, 37,  559,  4483,  21286,   67006,   164242, ...
  1, 46,  901,  9856,  67006,  304300,  1004590, ...
  1, 55, 1324, 18388, 164242, 1004590,  4443580, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 10,   1;
  1, 19,  19,    1;
  1, 28, 118,   28,    1;
  1, 37, 298,  298,   37,   1;
  1, 46, 559, 1540,  559,  46,  1;
  1, 55, 901, 4483, 4483, 901, 55, 1;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf.Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7).

Cf. A003683, A017173, A143680.

a143683 n k = a143683_tabl !! n !! k
a143683_row n = a143683_tabl !! n
a143683_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A246036 Expansion of (1+4x)/((1+2x)(1-4x)).

Original entry on oeis.org

1, 6, 20, 88, 336, 1376, 5440, 21888, 87296, 349696, 1397760, 5593088, 22368256, 89481216, 357908480, 1431666688, 5726601216, 22906535936, 91625881600, 366504050688, 1466015154176, 5864062713856, 23456246661120, 93824995033088, 375299963355136, 1501199886974976, 6004799480791040, 24019198057381888

Offset: 0

N. J. A. Sloane, Aug 21 2014

Also, fourth moments of Rudin-Shapiro polynomials (see Doche, Doche-Habsieger, Ekhad papers). - Doron Zeilberger, Apr 15 2016

Vincenzo Librandi, 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.
Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
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
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003683, A054881, A246037, A271494, A271495, A271496.

I:=[1,6]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 22 2014

CoefficientList[Series[(1+4x)/((1+2x)(1-4x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2014 *)

Vec((1+4*x)/((1+2*x)*(1-4*x)) + O(x^100)) \\ Colin Barker, Aug 22 2014

apply( A246036(n)=(4^(1+n)-(-2)^n)/3, [0..30]) \\ M. F. Hasler, Sep 18 2020

A246036= BinaryRecurrenceSequence(2,8,1,6)
[A246036(n) for n in range(41)] # G. C. Greubel, Mar 08 2023

a(n) = 2*a(n-1) + 8*a(n-2).
a(n) = (4^(1+n) - (-2)^n)/3. - Colin Barker, Aug 22 2014
a(n) = A054881(n+3)/8. - L. Edson Jeffery, Apr 22 2015
a(n) = A003683(n+2)/2 and the above formula follow from the explicit expression for a(n), cf. second formula. - M. F. Hasler, Sep 11 2020
a(n) = 2^n*A001045(n+2). - R. J. Mathar, Mar 08 2021

A071930 Number of words of length 2n in the two letters s and t that reduce to the identity 1 by using the relations ssTT=1, ststSS=1 and ststTT=1, where S and T are the inverses of s and t, respectively (i.e., sS=1 and tT=1). The generators s and t and the three stated relations generate the quaternion group Q4.

Original entry on oeis.org

0, 6, 12, 72, 240, 1056, 4032, 16512, 65280, 262656, 1047552, 4196352, 16773120, 67117056, 268419072, 1073774592, 4294901760, 17180000256, 68719214592, 274878431232, 1099510579200, 4398048608256, 17592181850112

Offset: 1

John W. Layman and Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com), Jun 14 2002

a(n) = A003683(n+1)/6. No words of odd length (see the description above) reduce to 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003674, A003683.

[4^(n-1)-(-2)^(n-1): n in [1..40]]; // G. C. Greubel, Feb 17 2023

Table[2^(2n-2)-(-2)^(n-1),{n,30}] (* or *) LinearRecurrence[{2,8},{0,6},30] (* Harvey P. Dale, Dec 10 2012 *)

[4^(n-1)-(-2)^(n-1) for n in range(1,41)] # G. C. Greubel, Feb 17 2023

a(n) = 2^(2*n-2) - (-2)^(n-1) = 6*A003683(n-1).
From Harvey P. Dale, Dec 10 2012: (Start)
a(n) = 2*a(n-1) + 8*a-(n-2).
G.f.: 6*x^2/(1-2*x-8*x^2). (End)
G.f.: Q(0), where Q(k)= 1 - 1/(4^k - 4*x*16^k/(4*x*4^k - 1/(1 + 1/(2*4^k - 16*x*16^k/(8*x*4^k +1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) = 2*A003674(n-1). - G. C. Greubel, Feb 17 2023

A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, -1, 1, 5, 3, -1, -2, 4, 10, 5, 0, -4, -4, 12, 20, 8, 1, -2, -13, -4, 31, 38, 13, 1, 3, -11, -33, 3, 73, 71, 21, 0, 6, 6, -42, -74, 34, 162, 130, 34, -1, 3, 24, 0, -130, -146, 128, 344, 235, 55, -1, -4, 21, 72, -50, -352

Offset: 0

Philippe Deléham, Nov 13 2006

			Triangle begins:
1;
1, 1;
0, 2, 2;
-1, 1, 5, 3;
-1, -2, 4, 10, 5;
0, -4, -4, 12, 20, 8;
1, -2, -13, -4, 31, 38, 13;
1, 3, -11, -33, 3, 73, 71, 21;
0, 6, 6, -42, -74, 34, 162, 130, 34;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)

Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).
T(n,0) = A010892(n).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A099254(n).
T(n+1,n) = A001629(n+2).
Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.
Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - Philippe Deléham, Dec 01 2006
T(n+2,n) = A129707(n+1).- Philippe Deléham, Dec 18 2011
G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - Philippe Deléham, Dec 18 2011

cp's OEIS Frontend

A084175 Jacobsthal oblong numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A091914 a(n) = 2*a(n-1) + 12*a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A143683 Pascal-(1,8,1) array.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Sage

Formula

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Views

A091914 a(n) = 2a(n-1) + 12a(n-2).

A246036 Expansion of (1+4x)/((1+2x)(1-4x)).