A024493 -id:A024493 - cp's OEIS frontend

A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

2, 3, 7, 13, 27, 53, 107, 213, 427, 853, 1707, 3413, 6827, 13653, 27307, 54613, 109227, 218453, 436907, 873813, 1747627, 3495253, 6990507, 13981013, 27962027, 55924053, 111848107, 223696213, 447392427, 894784853, 1789569707, 3579139413, 7158278827, 14316557653

Offset: 0

Michael Somos, Jun 17 1999

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

			G.f. = 2 + 3*x + 7*x^2 + 13*x^3 + 27*x^4 + 53*x^5 + 107*x^6 + 213*x^7 + 427*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, Journal de théorie des nombres de Bordeaux, Vol. 13, No. 1 (2001), pp. 27-41.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials, Ramanujan Journal 45.3 (2018): 659-681. See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers, Vol. 18 (2018), Article #A29.
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Sam Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, Vol. 117, No. 7 (2010), pp. 581-598.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 (first differences).
Cf. A001045, A014551, A024493, A062510.
Cf. A007283, A007679, A078008, A081254.
Cf. A083322, A083944, A083581, A084170.
Cf. A102713, A127978, A130755, A139763.
Cf. A140360, A140966, A166249, A168642.
Cf. A280560, A290604, A340627.
Cf. A112387, A135318.

[(5*2^n+(-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{1,2},{2,3},40] (* Harvey P. Dale, Dec 11 2017 *)

{a(n) = if( n<0, 0, (5*2^n + (-1)^n) / 3)};

{a(n) = if (n<0 ,0, if( n<2, n+2, a(n-1) + 2*a(n-2)))};

[(5*2^n+(-1)^n)/3 for n in range(35)] # G. C. Greubel, Apr 10 2019

G.f.: (2 + x) / (1 - x - 2*x^2).
a(n) = (5*2^n + (-1)^n) / 3.
a(n) = 2^(n+1) - A001045(n).
a(n) = A084170(n)+1 = abs(A083581(n)-3) = A081254(n+1) - A081254(n) = A084214(n+2)/2.
a(n) = 2*A001045(n+1) + A001045(n) (note that 2 is the limit of A001045(n+1)/A001045(n)). - Paul Barry, Sep 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=-charpoly(A,-1). - Milan Janjic, Jan 27 2010
Equivalently, with different offset, a(n) = b(n+1) with b(0)=1 and b(n) = Sum_{i=0..n-1} (-1)^i (1 + (-1)^i b(i)). - Olivier Gérard, Jul 30 2012
a(n) = A000975(n-2)*10 + 5 + 2*(-1)^(n-2), a(0)=2, a(1)=3. - Yuchun Ji, Mar 18 2019
a(n+1) = Sum_{i=0..n} a(i) + 1 + (1-(-1)^n)/2, a(0)=2. - Yuchun Ji, Apr 10 2019
a(n) = 2^n + J(n+1) = J(n+2) + J(n+1) - J(n), where J is A001045. - Yuchun Ji, Apr 10 2019
a(n) = A001045(n+2) + A078008(n) = A062510(n+1) - A078008(n+1) = (A001045(n+2) + A062510(n+1))/2 = A014551(n) + 2*A001045(n). - Paul Curtz, Jul 14 2021
From Thomas Scheuerle, Jul 14 2021: (Start)
a(n) = A083322(n) + A024493(n).
a(n) = A127978(n) - A102713(n).
a(n) = A130755(n) - A166249(n).
a(n) = A007679(n) + A139763(n).
a(n) = A168642(n) XOR A007283(n).
a(n) = A290604(n) + A083944(n). (End)
From Paul Curtz, Jul 21 2021: (Start)
a(n) = 5*A001045(n) - A280560(n+1) = abs(A140360(n+1)) - A280560(n+1).
a(n) = 2^n + A001045(n+1) = A001045(n+3) - A000079(n).
a(n) = A001045(n+4) - A340627(n). (End)
a(n) = A001045(n+5) - A005010(n).
a(n+1) + a(n) = a(n+2) - a(n) = 5*2^n. - Michael Somos, Feb 22 2023
a(n) = A135318(2*n) + A135318(2*n+1) = A112387(2*n) + A112387(2*n+1). - Paul Curtz, Jun 26 2024
E.g.f.: (cosh(x) + 5*cosh(2*x) - sinh(x) + 5*sinh(2*x))/3. - Stefano Spezia, May 18 2025

Formula of Milan Janjic moved here from wrong sequence by Paul D. Hanna, May 29 2010

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 4, 16, 1, 1, 1, 2, 8, 32, 1, 1, 1, 1, 5, 16, 64, 1, 1, 1, 1, 2, 11, 32, 128, 1, 1, 1, 1, 1, 6, 22, 64, 256, 1, 1, 1, 1, 1, 2, 16, 43, 128, 512, 1, 1, 1, 1, 1, 1, 7, 36, 85, 256, 1024, 1, 1, 1, 1, 1, 1, 2, 22, 72, 170, 512, 2048

Offset: 0

Seiichi Manyama, Mar 13 2019

			Square array begins:
     1,   1,  1,  1,  1,  1, 1, 1, 1, ...
     2,   1,  1,  1,  1,  1, 1, 1, 1, ...
     4,   2,  1,  1,  1,  1, 1, 1, 1, ...
     8,   4,  2,  1,  1,  1, 1, 1, 1, ...
    16,   8,  5,  2,  1,  1, 1, 1, 1, ...
    32,  16, 11,  6,  2,  1, 1, 1, 1, ...
    64,  32, 22, 16,  7,  2, 1, 1, 1, ...
   128,  64, 43, 36, 22,  8, 2, 1, 1, ...
   256, 128, 85, 72, 57, 29, 9, 2, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000079, A011782, A024493, A038503, A139398, A306847, A306852, A306859, A306860.

Cf. A306680.

T[n_, k_] := Sum[Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n,k*j).

A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.

Original entry on oeis.org

1, 1, 1, 2, 17, 101, 402, 1275, 3921, 14114, 58601, 243695, 950578, 3537847, 13166791, 50514102, 198627921, 782913717, 3054480306, 11824753551, 45823049817, 178682390994, 700285942731, 2747647985241, 10767833451954, 42164261091351, 165225573240651

Offset: 0

Paul Barry, May 16 2006

a(n) - A119364(n) = A119365(n).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Central coefficients of number triangle A119335.
a(n) = A119335(2n, n).
Cf. A024493, A119358, A139468, A139469.

Table[Sum[Binomial[n,3k]^2, {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Mar 12 2019 *)
Table[HypergeometricPFQ[{1/3 - n/3, 1/3 - n/3, 2/3 - n/3, 2/3 - n/3, -n/3, -n/3}, {1/3, 1/3, 2/3, 2/3, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 12 2019 *)

From Vaclav Kotesovec, Mar 12 2019: (Start)
Recurrence: (n-2)*(n-1)*n*(637*n^6 - 11466*n^5 + 84364*n^4 - 324394*n^3 + 686227*n^2 - 755060*n + 336132)*a(n) = 3*(n-2)*(n-1)*(1274*n^7 - 23569*n^6 + 180194*n^5 - 733383*n^4 + 1699606*n^3 - 2208294*n^2 + 1449504*n - 351000)*a(n-1) - 3*(n-2)*(3185*n^8 - 63700*n^7 + 539028*n^6 - 2512118*n^5 + 7020469*n^4 - 11971242*n^3 + 12050010*n^2 - 6446736*n + 1362744)*a(n-2) + (14014*n^9 - 315315*n^8 + 3072678*n^7 - 16986046*n^6 + 58535088*n^5 - 129861691*n^4 + 184326992*n^3 - 159830656*n^2 + 75517728*n - 14313456)*a(n-3) + 3*(n-3)*(3185*n^8 - 63700*n^7 + 538391*n^6 - 2501394*n^5 + 6946794*n^4 - 11707256*n^3 + 11530544*n^2 - 5915328*n + 1142208)*a(n-4) + 18*(n-4)*(n-3)*(2*n - 9)*(637*n^6 - 7644*n^5 + 36589*n^4 - 88858*n^3 + 114124*n^2 - 71840*n + 16440)*a(n-5).
a(n) ~ 4^n / (3*sqrt(Pi*n)). (End)

Edited by N. J. A. Sloane, Jun 12 2008

A130781 Sequence is identical to its third differences: a(n+3) = 3a(n+2) - 3a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

Original entry on oeis.org

1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530

Offset: 0

Paul Curtz, Jul 14 2007, Jul 18 2007

The inverse binomial transform is 1,0,1,... repeated with period 3, essentially A011655. - R. J. Mathar, Aug 28 2023

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

See A130750, A130752, A130755, A129339.

Essentially a duplicate of A024493.

a[n_] := a[n] = 3 a[n - 1] - 3 a[n - 2] + 2 a[n - 3]; a[0] = a[1] = 1; a[2] = 2; Table[a@n, {n, 0, 33}] (* Or *)
CoefficientList[ Series[(1 - 2 x + 2 x^2)/(1 - 3 x + 3 x^2 - 2 x^3), {x, 0, 33}], x] (* Robert G. Wilson v, Sep 08 2007 *)
LinearRecurrence[{3,-3,2},{1,1,2},40] (* Harvey P. Dale, Sep 17 2013 *)

3*a(n) = 2^(n+1) + A087204(n+1).
Also first differences of A024494.
G.f.: (1-2x+2x^2)/(1-3x+3x^2-2x^3).
Binomial transform of [1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...]; i.e., ones in positions 2, 5, 8, 11, ... and the rest zeros. [Corrected by Gary W. Adamson, Jan 07 2008]

Edited by N. J. A. Sloane, Jul 28 2007

A091917 Coefficient array of polynomials (z-1)^n-1.

Original entry on oeis.org

1, -2, 1, 0, -2, 1, -2, 3, -3, 1, 0, -4, 6, -4, 1, -2, 5, -10, 10, -5, 1, 0, -6, 15, -20, 15, -6, 1, -2, 7, -21, 35, -35, 21, -7, 1, 0, -8, 28, -56, 70, -56, 28, -8, 1, -2, 9, -36, 84, -126, 126, -84, 36, -9, 1, 0, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1

Offset: 0

Paul Barry, Feb 13 2004

The first element has been changed to 1 to produce an invertible matrix. Alternatively, this is the coefficient array for the polynomials P(z,n) = Product_{j=0..n-1} (z-(1+w(n)^j)) where w(n) = e^(2*Pi*i/n), i=sqrt(-1).
The row entries determine interesting recurrences. For instance, a(n) = 4a(n-1) + 6a(n-2) + 4a(n-3), a(0)=a(1)=a(2)=1, gives A038503. Sequences of the form a(n) = Sum_{k=0..n} (binomial(n,k) if k mod m = r, otherwise 0), for r=0..m-1, result. Equivalently, a(n) = Sum_{j=0..n-1} 2^n*(cos(Pi*j/m))^n*cos((n-2r)Pi*j/m)/m, r=0..m-1. These include A024493, A024494, A024495, A038503, A038504, A038505. The inverse matrix is A091918.
Triangle T(n,k), 0 <= k <= n, read by rows given by [ -2, 2, 1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 11 2007

			Rows begin:
  { 1},
  {-2,  1},
  { 0, -2,  1},
  {-2,  3, -3,  1},
  { 0, -4,  6, -4,  1},
  ...

Alois P. Heinz, Rows n = 0..140, flattened

T:= n-> `if`(n=0, 1, (p-> seq(coeff(p,z,i), i=0..n))((z-1)^n-1)):
seq(T(n), n=0..12);  # Alois P. Heinz, May 23 2015

Table[If[n == 0, 1, CoefficientList[(z-1)^n-1, z]], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)

row(n) = if (n==0, 1, Vecrev((z-1)^n-1)); \\ Michel Marcus, May 23 2015

T(n,k) = T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = -2, T(2,0) = 0, T(n,k) = 0 for k > n or for k < 0. - Philippe Deléham, May 23 2015

G.f.: (1-2*x-x^2+x^2*y)/((x-1)*(-x+x*y-1)). - R. J. Mathar, Aug 11 2015

A097122 Expansion of (1-x)^2/((1-x)^3 - 3*x^3).

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 70, 169, 421, 1036, 2521, 6139, 14998, 36661, 89545, 218644, 533941, 1304071, 3184966, 7778449, 18996733, 46394716, 113307745, 276726019, 675833686, 1650553981, 4031064961, 9844867684, 24043624093, 58720529071

Offset: 0

Paul Barry, Jul 25 2004

Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,4).

Cf. A024493, A052101, A100136.

CoefficientList[Series[(1-x)^2/((1-x)^3-3x^3),{x,0,40}],x]

a(n) = sum(k=0, n\3, binomial(n, 3*k) * 3^k); \\ Michel Marcus, Oct 11 2021

G.f.: (1-2*x+x^2)/(1-3*x+3*x^2-4*x^3).
a(n) = 3*a(n-1) - 3*a(n-2) + 4*a(n-3).
a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3k) * 3^k.

A100137 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-6k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 65, 136, 296, 672, 1584, 3840, 9473, 23566, 58736, 146080, 361760, 891328, 2184961, 5331476, 12958684, 31400160, 75910320, 183220800, 441787201, 1064687642, 2565404524, 6181873208, 14899796416, 35922756992, 86635757825

Offset: 0

Paul Barry, Nov 06 2004

Binomial transform of 1,1,1,1,1,1,2,2,2,5,5,11,11,... with g.f. (1-x)^2(1+x)^2/(1-3x^2+3x^4-2x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-x^6).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8,0,0,1).

Cf. A024493, A100131, A100134, A100137, A100138.

Table[Sum[Binomial[n-3k,3k]2^(n-6k),{k,0,Floor[n/6]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,0,0,1},{1,2,4,8,16,32},31] (* Harvey P. Dale, Mar 19 2015 *)

G.f.: (1-2x)^2/((1-2x)^3 - x^6).

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + a(n-6).

A131090 First differences of A131666.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 7, 15, 28, 57, 113, 228, 455, 911, 1820, 3641, 7281, 14564, 29127, 58255, 116508, 233017, 466033, 932068, 1864135, 3728271, 7456540, 14913081, 29826161, 59652324, 119304647, 238609295, 477218588, 954437177, 1908874353

Offset: 0

Paul Curtz, Sep 24 2007

The first differences b(n)=a(n+1)-a(n) obey the recurrence b(n+1)-2b(n) = (-3,3,-2,3,-3,2), continued with period 6.
The 2nd differences c(n)=b(n+1)-b(n) obey the recurrence c(n+1)-2c(n) = (6,-5,5,-6,5,-5), periodically continued with period 6.
The hexaperiodic coefficients in these recurrences for A113405, A131666 and their higher order differences define a table,
0, 0, 1, 0, 0, -1 <- A113405
0, 1, -1, 0, -1, 1 <- A131666
1, -2, 1, -1, 2, -1 <- a(n)
-3, 3, -2, 3, -3, 2 <- b(n)
6, -5, 5, -6, 5, -5 <- c(n)
-11,10,-11, 11,-10, 11
21,-21,22,-21, 21,-22
...
in which the first three columns are A024495, A131708 and A024493, multiplied by a checkerboard pattern of signs.

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

LinearRecurrence[{2,0,-1,2},{0,1,0,1},40] (* Harvey P. Dale, Jan 15 2016 *)

a(n) = A131666(n+1)-A131666(n).
a(n+1)-2a(n) = A131556(n), a sequence with period length 6.
G.f.: -(x-1)^2*x / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, Mar 04 2013

Edited by R. J. Mathar, Jun 28 2008

A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, -1, -2, -3, 0, 0, 1, 3, 6, -1, -1, -1, -2, -5, -11, 0, 1, 2, 3, 5, 10, 21, 0, 0, -1, -3, -6, -11, -21, -42, 1, 1, 1, 2, 5, 11, 22, 43, 85, 0, -1, -2, -3, -5, -10, -21, -43, -86, -171, 0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, -1, -1, -1, -2, -5, -11, -22, -43, -85, -170, -341, -683, 0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365

Offset: 0

Paul Curtz, Nov 07 2009

The array contains A131708(0) in diagonal 0, then -A024495(0..1) in diagonal 1, then A024493(0..2) in diagonal 2, then -A131708(0..3), then A024495(0..4), then -A024493(0..5).

			The table starts in row n=0 with columns k >= 0 as:
0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0 A131531
0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1 A092220
1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2 A131556
-3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3 A164359
6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5
-11, 10, -11, 11, -10, 11, -11, 10, -11, 11, -10, 11, -11, 10, -11
21, -21, 22, -21, 21, -22, 21, -21, 22, -21, 21, -22, 21, -21, 22

Cf. A167617 (antidiagonal sums).

A131531 := proc(n) op((n mod 6)+1,[0,0,1,0,0,-1]) ; end proc:
A167613 := proc(n,k) option remember; if n= 0 then A131531(k); else procname(n-1,k+1)-procname(n-1,k) ; end if;end proc: # R. J. Mathar, Dec 17 2010

nmax = 13;
A131531 = Table[{0, 0, 1, 0, 0, -1}, {nmax}] // Flatten;
T[n_] := T[n] = Differences[A131531, n];
T[n_, k_] := T[n][[k]];
Table[T[n-k, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 20 2023 *)

T(0,k) = A131531(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.
T(n,n) = A001045(n). T(n,n+1) = -A001045(n). T(n,n+2) = A078008(n).
T(n,0) = -T(n,3) = (-1)^(n+1)*A024495(n).
T(n,1) = (-1)^(n+1)*A131708(n).
T(n,2) = (-1)^n*A024493(n).
T(n,k+6) = T(n,k).
a(n) = A131708(0), -A024495(0,1), A024493(0,1,2), -A131708(0,1,2,3), A024495(0,1,2,3,4), -A024493(0,1,2,3,4,5).

A094715 a(n) = Sum_{2i+3j=n, 0<=i<=n, 0<=j<=n} n!/( (2i)!(3*j)! ).

Original entry on oeis.org

1, 0, 1, 1, 1, 10, 2, 35, 29, 85, 211, 220, 926, 1001, 3095, 5461, 9829, 25126, 37130, 97223, 164921, 349525, 728575, 1309528, 2973350, 5326685, 11450531, 22369621, 43942081, 91869970, 174174002, 365088395, 708653429, 1431655765, 2884834891

Offset: 0

Benoit Cloitre, May 23 2004

nonn

Average of binomial and inverse binomial transform of {1, 0, 0, 1, 0, 0, 1, ...}. - Paul Barry, Jan 04 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000748, A010892, A024493, A094717.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) )); // G. C. Greubel, Feb 13 2023

A094715_list := proc(n) local i; (exp(z)+2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3;  series(%,z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A094715_list(34); # Peter Luschny, Jul 10 2012

Table[(1/6)*(Boole[n==0] +2^n +2*ChebyshevU[n,1/2] -ChebyshevU[n-1, 1/2] +2*3^(n/2)*ChebyshevU[n, -Sqrt[3]/2] +3^((n+1)/2)*ChebyshevU[n- 1, -Sqrt[3]/2]), {n,0,50}] (* G. C. Greubel, Feb 13 2023 *)

a(n)=sum(i=0,n,sum(j=0,n,if(n-2*i-3*j,0,n!/(2*i)!/(3*j)!)))

def A094715_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) ).list()
A094715_list(50) # G. C. Greubel, Feb 13 2023

Limit_{n --> oo} a(n)/2^n = 1/6.
G.f.: (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)). - Vladeta Jovovic, May 23 2004
a(n) = (1/3)*Sum_{k=0..floor(n/2)} C(n, 2*k)*(2*cos(2*Pi*(n-2*k)/3) + 1). - Paul Barry, Jan 04 2005 [corrected by Jason Yuen, Aug 28 2024]
E.g.f.: (exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3. - Peter Luschny, Jul 10 2012
a(n) = (1/6)*([n=0] + 2^n + 2*A010892(n) - A010892(n-1) + 2*A000748(n) + 3*A000748(n-1)). - G. C. Greubel, Feb 13 2023

cp's OEIS Frontend

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A094715 a(n) = Sum_{2i+3j=n, 0<=i<=n, 0<=j<=n} n!/( (2i)!(3*j)! ).