A009545 -id:A009545 - cp's OEIS frontend

A291482 Expansion of e.g.f. arcsin(x)*exp(x).

0, 1, 2, 4, 8, 24, 80, 456, 2368, 20352, 139648, 1577984, 13327360, 185992832, 1860708096, 30882985472, 356724338688, 6860887896064, 89815091306496, 1963843714723840, 28724760194564096, 703639672161697792, 11370790299166343168, 308435832182144040960, 5456591088206554333184, 162354575283061816197120

Offset: 0

Ilya Gutkovskiy, Aug 24 2017

nonn

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 24*x^5/5! + ...

Cf. A001818, A009545, A012316, A081919 (first differences).

a:=series(arcsin(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019

nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSin[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Exp[x] x Sqrt[1 - x^2]/(1 + ContinuedFractionK[-2 x^2 Floor[(k + 1)/2] (2 Floor[(k + 1)/2] - 1), 2 k + 1, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Sum[(x^(2 k + 1) Pochhammer[1/2, k])/(k! + 2 k k!), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n,2k+1]Binomial[2k,k] (2k)!/4^k,{k,0,(n-1)/2}],{n,0,12}] (* Emanuele Munarini, Dec 17 2017 *)

makelist(sum(binomial(n,2*k+1)*binomial(2*k,k)*(2*k)!/4^k,k,0,floor((n-1)/2)),n,0,12); /* Emanuele Munarini, Dec 17 2017 */

x='x+O('x^99); concat(0, Vec(serlaplace(asin(x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

E.g.f.: exp(x)*x*sqrt(1 - x^2)/(1 - 1*2*x^2/(3 - 1*2*x^2/(5 - 3*4*x^2/(7 - 3*4*x^2/(9 - ...))))), a continued fraction.
a(n) ~ (exp(2) - (-1)^n) * n^(n-1) / exp(n+1). - Vaclav Kotesovec, Aug 26 2017
From Emanuele Munarini, Dec 17 2017: (Start)
a(n) = Sum_{k=0..(n-1)/2} binomial(n,2*k+1)*binomial(2*k,k)* (2k)!/4^k.
a(n+4) - 2*a(n+3) - (n^2+4*n+3)*a(n+2) + (n+2)*(2*n+3)*a(n+1) - (n+1)*(n+2)*a(n) = 0. (End)

A348691 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348690 gives the real part.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 1, -1, 2, -2, -1, -1, 0, -2, -1, 1, 0, -4, -3, 1, -2, 0, 1, 3, -2, -2, -1, 3, 0, 2, 3, 1, -4, -4, -3, 5, -2, 4, 5, 3, -2, 2, 3, 3, 4, 2, 3, -3, -4, 0, 1, 5, 2, 4, 5, -1, 2, 2, 3, -1, 4, -2, -1, -3, -8, 0, 1, 9, 2, 8, 9, -1, 2, 6, 7, -1, 8, -2

Offset: 0

Rémy Sigrist, Oct 29 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The function f has similarities with A065620; here the nonzero digits in base 1+i cycle through powers of i, there nonzero digits in base 2 cycle through powers of -1.
If we replace 1's in binary expansions by powers of i from left to right (rather than right to left as here), then we obtain the Lévy C curve (A332251, A332252).

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Chandler Davis and Donald Knuth, Number representations and Dragon Curves I, Journal of Recreational Mathematics, volume 3, number 2 (April 1970), pages 66-81. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2011, pages 571-614.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 4)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))

See A332251, A332252 for a similar sequence.

Cf. A000120, A009545, A065620, A070939, A348690.

a(n) = { my (v=0, o=0, x); while (n, n-=2^x=valuation(n, 2); v+=I^o * (1+I)^x; o++); imag(v) }

a(2^k) = A009545(k) for any k >= 0.

A144081 Eigentriangle generated from expansion of sin(x)*exp(x), row sums = (2^n - 1).

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 0, 2, 6, 7, -4, 0, 6, 14, 15, -8, -4, 0, 14, 30, 31, -8, -8, -12, 0, 30, 62, 63, 0, -8, -24, -28, 0, 62, 126, 127, 16, 0, -24, -56, -60, 0, 126, 254, 255, 32, 16, 0, -56, -120, -124, 0, 254, 510, 511, 32, 32, 48, 0, -120, -248, -252, 0, 510, 1022, 1023

Offset: 1

Gary W. Adamson, Sep 10 2008

Row sums = (2^n - 1): (1, 3, 7, 15, 31,...) = INVERT transform of A009545 starting with offset 1. Right border = (1, 1, 3, 7, 15,...).
Left border = A009545, = expansion of sin(x)*exp(x) starting with offset 1.
Sum of row n terms = rightmost term of next row.

			First few rows of the triangle =
   1;
   2,  1;
   2,  2,   3;
   0,  2,   6,   7;
  -4,  0,   6,  14,  15;
  -8, -4,   0,  14,  30, 31;
  -8, -8, -12,   0,  30, 62,  63;
   0, -8, -24, -28,   0, 62, 126, 127;
  16,  0, -24, -56, -60,  0, 126, 254, 255;
  ...
Row 4 = (0, 2, 6, 7) pairwise product of (0, 2, 2, 1) and (1, 1, 3, 7) = (0*1, 2*1, 2*3, 1*7); where (1, 2, 2, 0,...) = the first 4 terms of A009545 with offset 1.

Cf. A000225, A009545.

a25(n) = if (n, 2^n-1, 1); \\ A000225
a45(n) = (1+I)^(n-2) + (1-I)^(n-2); \\ A009545
T(n,k) = if (n>=k, a45(n-k+1)*a25(k-1), 0);
row(n) = vector(n, k, a45(n-k+1)*a25(k-1)); \\ Michel Marcus, Nov 20 2022

T(n,k) = A009545(n-k+1)*A000225(k-1).
A009545 = expansion of sin(x)*exp(x), starting with offset 1: (1, 2, 2, 0, -4, -8, -8,...).
A000225(k-1) = A000225 offset: (1, 1, 3, 7, 15, 31, 63, 127,...).
These operations = the following: Matrix A = an infinite lower triangular matrix with rows = A009545 subsequences decrescendo: (1; 2,1; 2,2,1; 0,2,2,1; -4,0,2,2,1;...) and matrix B = an infinite lower triangular matrix with (1, 1, 3, 7, 15,...) in the main diagonal and the rest zeros.
This triangle = A*B.

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

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

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A206306 Riordan array (1, x/(1-3x+2x^2)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1

Offset: 0

Philippe Deléham, Feb 06 2012

The convolution triangle of the Mersenne numbers A000225. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,    1;
  0,    3,    1;
  0,    7,    6,     1;
  0,   15,   23,     9,     1;
  0,   31,   72,    48,    12,     1;
  0,   63,  201,   198,    82,    15,    1;
  0,  127,  522,   699,   420,   125,   18,    1;
  0,  255, 1291,  2223,  1795,   765,  177,   21,   1;
  0,  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27,  1;

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

Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
Cf. A008585, A009545, A084938, A062725, A110441, A204089, A204091.
Cf. A045741, A078020, A244038.

function T(n,k) // T = A206306
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return 0;
  else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022

# Uses function PMatrix from A357368.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)

def T(n,k): # T = A206306
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022

Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonals sums are even-indexed Fibonacci numbers.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
From Philippe Deléham, Nov 17 2013; corrected Feb 13 2020: (Start)
T(n, n) = 1.
T(n+1, n) = 3n = A008585(n).
T(n+2, n) = A062725(n).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
From G. C. Greubel, Dec 20 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
T(2*n, n+1) = A045741(n+2), n >= 0.
T(2*n+1, n+1) = A244038(n). (End)

A217988 Binomial transform of A215495(n).

Original entry on oeis.org

1, 2, 4, 10, 26, 66, 160, 372, 840, 1864, 4096, 8944, 19424, 41952, 90112, 192576, 409728, 868480, 1835008, 3866368, 8125952, 17038848, 35651584, 74449920, 155191296, 322963456, 671088640, 1392504832, 2885672960, 5972680704, 12348030976, 25501384704

Offset: 0

Paul Curtz, Oct 17 2012

Companion to A218009.
Like any other sequence with a linear recurrence with constant coefficients, this sequence is periodic if read modulo some constant m. These Pisano period lengths for m>=1 are 1, 1, 8, 1, 20, 8, 168, 1, 24, 20, 440, 8, 156, 168, 40, 1, 272, 24, 1368, 20, ... [Curtz's comment reformulated and extended by R. J. Mathar, Oct 23 2012]
Let b(n) = a(n+1)-2*a(n), then b(n+3)-2*b(n+2) = A009545(n+2). - edited by Michel Marcus, Apr 24 2018

			a(n) and successive differences:
1, 2,  4, 10, 26,  66, 160, 372,  840, 1864, 4096, ...
1, 2,  6, 16, 40,  94, 212, 468, 1024, ...
1, 4, 10, 24, 54, 118, 256, ...
3, 6, 14, 30, 64, ...
3, 8, 16, ...
5, 8, ...
3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Cf. A214282, A214283.

I:=[1, 2, 4, 10, 26, 66]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012

a[n_] := Sum[ Binomial[n, k]*If[ OddQ[k], k, k/2 + Boole[ Mod[k, 4] == 0]], {k, 0, n}]; Table[ a[n], {n, 0, 31}] (* Jean-François Alcover, Oct 17 2012 *)
CoefficientList[Series[(1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2 * (1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{6,-14,16,-8},{1,2,4,10,26,66},40] (* Harvey P. Dale, Aug 14 2018 *)

x='x+O('x^30); Vec((1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2))) \\ G. C. Greubel, Apr 23 2018

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4) with n > 5.
a(n) = A218009(n) + A146559(n).
G.f.: (1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 22 2012
a(n) = 2^(n-3)*(3*n+2)+((1+i)^n+(1-i)^n)/4, where i=sqrt(-1) and n>1, with a(0)=1, a(1)=2.

A218009 Binomial transform of A212831(n).

Original entry on oeis.org

0, 1, 4, 12, 30, 70, 160, 364, 824, 1848, 4096, 8976, 19488, 42016, 90112, 192448, 409472, 868224, 1835008, 3866880, 8126976, 17039872, 35651584, 74447872, 155187200, 322959360, 671088640, 1392513024, 2885689344, 5972697088, 12348030976, 25501351936, 52613316608

Offset: 0

Paul Curtz, Oct 18 2012

Companion to A217988.
Considering a(n+1) - 2*a(n) = 1,2,4,6,10,20,44,96,200,... = b(n), is
b(n+3) - 2*b(n+2) = -2,-2,0,4,8,8,0,-16,-32,-32,0,... = -A009545(n+2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Cf. A212831, A214282, A214283, A217988, A009545.

I:=[0, 1, 4, 12, 30, 70]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012

a[n_] := Sum[(1/4)*Binomial[n, k]*((-(1 + (-1)^k))*(-1 + (-1)^Floor[k/2]) - (-3 + (-1)^k)*k), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 18 2012 *)
CoefficientList[Series[x*(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4)/((1 - 2*x)^2*(1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)

G.f.: x*(1-2*x+2*x^2-2*x^3+2*x^4)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 18 2012
a(n) = 2^(n-3)*(3*n+2)+(1+i)^(n-4)+(1-i)^(n-4), where i=sqrt(-1) and n>1, with a(0)=0, a(1)=1. - Bruno Berselli, Oct 18 2012
a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -8*a(n-4).

A291483 Expansion of e.g.f. arcsinh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 2, 0, 4, 40, -64, -1344, 3984, 85408, -356896, -8462080, 45908160, 1209040768, -8080805888, -235449260032, 1871655631104, 59955521585664, -552758145525248, -19339870285225984, 202927333558572032, 7707208199780517888, -90698934927786770432, -3718489569130941169664, 48507735629457304555520

Offset: 0

Ilya Gutkovskiy, Aug 24 2017

sign

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 2*x^3/3! + 4*x^5/5! + 40*x^6/6! - 64*x^7/7! - 1344*x^8/8! + ...

Cf. A001818, A009545, A012584, A131577, A291482.

a:=series(arcsinh(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019

nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSinh[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Log[x + Sqrt[1 + x^2]] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[-Sum[((-1)^k (-1 + x + Sqrt[1 + x^2])^k)/k, {k, 1, Infinity}] Exp[x], {x, 0, nmax}], x]

E.g.f.: log(x + sqrt(1 + x^2))*exp(x).

A348761 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} ((-1)^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348760 gives the real part.

Original entry on oeis.org

0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, -4, 4, 5, -5, 6, -6, -5, 5, 6, -6, -5, 5, -4, 4, 5, -5, 4, -4, -3, 3, -2, 2, 3, -3, -2, 2, 3, -3, 4, -4, -3, 3, -8, 8, 9, -9, 10, -10, -9, 9, 10, -10

Offset: 0

Rémy Sigrist, Oct 31 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored representation of f(n) for n < 2^20 in the complex plane (the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 2)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n), darker shades correspond to higher values)

See A348691 for a similar sequence.

Cf. A000120, A009545, A070939, A348760.

a(n) = { my (v=0, k, o=-1); while (n, n-=2^k=valuation(n,2); v+=(1+I)^k * (-1)^o++); imag(v) }

a(2^k) = A009545(k) for any k >= 0.

A373358 a(n) = 4a(n-1) -5a(n-2) +2a(n-3) +2a(n-4) for a(0) = a(1) = 0, a(2) = 1, a(3) = 4 for n >= 4.

Original entry on oeis.org

0, 0, 1, 4, 11, 26, 59, 136, 323, 782, 1903, 4620, 11175, 26970, 65051, 156944, 378811, 914566, 2208199, 5331476, 12871663, 31074802, 75020243, 181113240, 437244675, 1055602590, 2548453951, 6152518684, 14853499511, 35859517706, 86572518539, 209004522016, 504581529803, 1218167581622

Offset: 0

Paul Curtz, Jun 02 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2).

Cf. A000129, A009545, A077893, A077953, A077980, A114203, A146559, A373245.

LinearRecurrence[{4, -5, 2, 2}, {0, 0, 1, 4}, 50] (* Paolo Xausa, Jun 19 2024 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,4d-5c+2b+2a}; NestList[nxt,{0,0,1,4},40][[;;,1]] (* Harvey P. Dale, Jan 11 2025 *)

a(n) = ((([2, 1; 1, 0]^(n+1))[2, 1]) - (1+I)^(n-1) - (1-I)^(n-1))/3 \\ Thomas Scheuerle, Jun 03 2024

G.f.: x^2 / ( (1 - 2*x - x^2) * (1 - 2*x + 2*x^2) ).
E.g.f.: exp(x)*(2*cosh(sqrt(2)*x) - 2*(cos(x)+sin(x)) + sqrt(2)*sinh(sqrt(2)*x))/6.
a(n) = A373245(n+1) - A114203(n+1).
a(0) = 0, a(n) = A373245(n-1) + A146559(n-1).
Binomial transform of 0, 0, followed by A077893 = abs(A077953) = abs(A077980).
a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) +2*a(n-4) for n >= 4.
From Thomas Scheuerle, Jun 03 2024: (Start)
a(n) = (A000129(n+1) - A009545(n+1))/3.
a(n) = (-i*sqrt(2)*(1-i)^(n+1) + i*sqrt(2)*(1+i)^(n+1) - (1-sqrt(2))^(n+1) + (1+sqrt(2))^(n+1))/(6*sqrt(2)).
a(n) = 2^n*(hypergeom([1/2 - n/2, -n/2], [-n], -1) - hypergeom([1/2 - n/2, -n/2], [-n], 2))/3. (End)

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A373358 a(n) = 4a(n-1) -5a(n-2) +2a(n-3) +2a(n-4) for a(0) = a(1) = 0, a(2) = 1, a(3) = 4 for n >= 4.