A146559 -id:A146559 - cp's OEIS frontend

A077860 Expansion of 1/((1 - 2x + 2x^2)*(1-x)).

1, 3, 5, 5, 1, -7, -15, -15, 1, 33, 65, 65, 1, -127, -255, -255, 1, 513, 1025, 1025, 1, -2047, -4095, -4095, 1, 8193, 16385, 16385, 1, -32767, -65535, -65535, 1, 131073, 262145, 262145, 1, -524287, -1048575, -1048575, 1, 2097153, 4194305, 4194305, 1, -8388607, -16777215, -16777215

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,-4,2).

I:=[1,3,5]; [n le 3 select I[n] else 3*Self(n-1)-4*Self(n-2)+2*Self(n-3): n in [1..60]]; // Vincenzo Librandi, Jul 02 2015

Join[{a=1,b=3},Table[c=2*b-2*a+1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
CoefficientList[Series[1/((1-2x+2x^2)(1-x)),{x,0,60}],x] (* or *) LinearRecurrence[{3,-4,2},{1,3,5},60] (* Harvey P. Dale, Feb 01 2013 *)

a(n):=sum((-1)^k*2^(n-k)*binomial(n-k-1,k),k,0,n); /* Vladimir Kruchinin, Jul 02 2015 */

Vec(1/((1-2*x+2*x^2)*(1-x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 1 + 2*imag((1 + I)^n); \\ Daniel Suteu, Dec 21 2018

a(n) = 1-A146559(n+2). a(n)= 3*a(n-1) -4*a(n-2) +2*a(n-3). - R. J. Mathar, Jan 18 2011
G.f.: Q(0) where Q(k) = 1 + k*(2*x+1) + 8*x - 2*x*(k+1)*(k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
G.f.: G(0)/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = Sum_{k=0..n} ((-1)^k*2^(n-k)*binomial(n-k-1,k)). - Vladimir Kruchinin, Jul 02 2015
a(n) = 1 + 2^(1 + n/2)*sin((n*Pi)/4). - Jean-François Alcover, Jul 02 2015
a(n) = 1 + 2*Im((1 + i)^n), where i is the imaginary unit. - Daniel Suteu, Dec 21 2018
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n+2,2*k+2). - Taras Goy, Jan 03 2025
E.g.f.: exp(x)*(1 + 2*sin(x)). - Stefano Spezia, Jan 03 2025

A121625 Real part of (n + n*i)^n.

Original entry on oeis.org

1, 1, 0, -54, -1024, -12500, 0, 6588344, 268435456, 6198727824, 0, -9129973459552, -570630428688384, -19384006821904192, 0, 56050417968750000000, 4722366482869645213696, 211773507042902211629312, 0, -1012950863698080557631477248, -107374182400000000000000000000

Offset: 0

Gary W. Adamson, Aug 12 2006

sign

			a(7) = 6588344 since (7 + 7i)^7 = (6588344 - 6588344i).

Cf. A115415, A121626, A146559.

a[n_] := Re[(n + n*I)^n]; Array[a, 19] (* Robert G. Wilson v, Aug 17 2006 *)

a(n) = real((n + n*I)^n); \\ Michel Marcus, Dec 19 2020

def A121625(n): return n**n*((1, 1, 0, -2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

a(n) = Re(n + n*i)^n.
From Chai Wah Wu, Feb 15 2024: (Start)
a(n) = n^n*Re((1+i)^n) = n^n*A146559(n) = n^n*Sum_{n=0..floor(n/2)} binomial(n,2j)*(-1)^j.
a(n) = 0 if and only if n==2 mod 4, as (1+i)^2=2i is purely imaginary, (1+i)^4=-4 is a nonzero real and (1+i) and (1+i)^3=-2+2i both have nonzero real parts.
(End)

More terms from Robert G. Wilson v, Aug 17 2006

a(0)=1 prepended by Alois P. Heinz, Dec 19 2020

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A193410 Expansion of (1-3x)/(1-6x+18*x^2).

Original entry on oeis.org

1, 3, 0, -54, -324, -972, 0, 17496, 104976, 314928, 0, -5668704, -34012224, -102036672, 0, 1836660096, 11019960576, 33059881728, 0, -595077871104, -3570467226624, -10711401679872, 0, 192805230237696, 1156831381426176, 3470494144278528

Offset: 0

Bruno Berselli, Aug 04 2011

Also real parts of 3^n*(1+i)^n, where i=sqrt(-1).

If |a(n)| > 0 then it is in A130505.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-18).

Cf. A066771, A121622, A130505.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)/(1-6*x+18*x^2))); /* or */ &cat[[r,3*r,0,-54*r] where r is (-324)^n: n in [0..6]];

I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)-18*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 26 2013

CoefficientList[Series[(1 - 3 x)/(1 - 6 x + 18 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{6,-18},{1,3},40] (* Harvey P. Dale, Jul 27 2021 *)

makelist(coeff(taylor((1-3*x)/(1-6*x+18*x^2), x, 0, n), x, n), n, 0, 25);

PARI
```
Vec((1-3*x)/(1-6*x+18*x^2) +O(x^26))
```

G.f.: (1-3*x)/(1-6*x+18*x^2).
a(n) = 3^n*A146559(n) = (1/2)*((3+3*i)^n+(3-3*i)^n), where i=sqrt(-1).
a(n) = 6*a(n-1)-18*a(n-2) for n>1.
a(n) = (3*sqrt(2))^n*cos(pi*n/4).
a(4k+2) = 0, a(4k+1) = 3*a(4k) = 18*a(4k-1) = 3*(-324)^k.
G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(3*k+3)/(x*(3*k+6) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

A348690 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the real 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 A348691 gives the imaginary part.

Original entry on oeis.org

0, 1, 1, 0, 0, -1, -1, 0, -2, -1, -1, 2, -2, 1, 1, 2, -4, 1, 1, 4, 0, 3, 3, 0, -2, 3, 3, 2, 2, 1, 1, -2, -4, 5, 5, 4, 4, 3, 3, -4, 2, 3, 3, -2, 2, -3, -3, -2, 0, 5, 5, 0, 4, -1, -1, -4, 2, -1, -1, -2, -2, -3, -3, 2, 0, 9, 9, 0, 8, -1, -1, -8, 6, -1, -1, -6, -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 for a similar sequence.

Cf. A000120, A065620, A146559, A348691, A332251, A332252.

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

a(2^k) = A146559(k) for any k >= 0.

A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 13, 9, 1, 0, 1, 46, 56, 16, 1, 0, 1, 199, 334, 160, 25, 1, 0, 1, 1072, 2157, 1408, 365, 36, 1, 0, 1, 6985, 15701, 12445, 4417, 721, 49, 1, 0, 1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0, 1, 462331, 1191336, 1183216, 597026, 166716, 25956, 2136, 81, 1, 0

Offset: 0

Philippe Deléham, Nov 19 2011

Row sums : A000142(n) = n!.

			Triangle begins :
1
1, 0
1, 1, 0
1, 4, 1, 0
1, 13, 9, 1, 0
1, 46, 56, 16, 1, 0
1, 199, 334, 160, 25, 1, 0
1, 1072, 2157, 1408, 365, 36, 1, 0
1, 6985, 15701, 12445, 4417, 721, 49, 1, 0
1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0

Alois P. Heinz, Rows n = 0..140, flattened
Sergey Kitaev, Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

Cf. A000290, A014145,

DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k + 1]] + y*s[[k + 1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k - 1] + q[k]*p[n - 1, k + 1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n - k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
m = 10;
DELTA[LinearRecurrence[{1, 1, -1}, {1, 0, 2}, m], LinearRecurrence[{0, 1}, {0, 1}, m], m] // Flatten (* Jean-François Alcover, Feb 21 2019 *)

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A172485(n+1), A146559(n), A000012(n), A000142(n), A003319(n), A111529(n), A111530(n), A111531(n), A111532(n), A111533(n) for x = -2,-1,0,1,2,3,4,5,6,7 respectively.
T(k+2,k)=(k+1)^2 = A000290(k+1).
T(n+1,1)= A014145(n).

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.

A348760 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the real 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 A348761 gives the imaginary part.

Original entry on oeis.org

0, 1, 1, 0, 0, 1, 1, 0, -2, 3, 3, -2, 2, -1, -1, 2, -4, 5, 5, -4, 4, -3, -3, 4, 2, -1, -1, 2, -2, 3, 3, -2, -4, 5, 5, -4, 4, -3, -3, 4, 2, -1, -1, 2, -2, 3, 3, -2, 0, 1, 1, 0, 0, 1, 1, 0, -2, 3, 3, -2, 2, -1, -1, 2, 0, 1, 1, 0, 0, 1, 1, 0, -2, 3, 3, -2, 2, -1

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 A348690 for a similar sequence.

Cf. A000120, A070939, A146559, A348761.

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

a(2^k) = A146559(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)

A102486 a(n) = 4a(n-1) - 5a(n-2).

Original entry on oeis.org

2, 6, 14, 26, 34, 6, -146, -614, -1726, -3834, -6706, -7654, 2914, 49926, 185134, 490906, 1037954, 1697286, 1599374, -2088934, -16352606, -54965754, -138099986, -277571174, -419784766, -291283194, 933791054, 5191580186, 16097365474, 38431560966, 73239416494, 100799861146

Offset: 0

N. J. A. Sloane, Feb 25 2005

Inverse binomial transform is 2,4,4,0,-8,-16,-16,.. essentially -A146559(n+3). - R. J. Mathar, Apr 07 2022

B. M. E. Moret and H. D. Shapiro, Algorithms from P to NP, Benjamin/Cummings, Vol. 1, 1991; p. 65.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5).

Cf. A099456.

I:=[2, 6]; [n le 2 select I[n] else 4*Self(n-1)-5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jan 15 2012

a := proc(n) option remember; if n = 0 then RETURN(2) end if; if n = 1 then RETURN(6) end if; 4*a(n - 1) - 5*a(n - 2); end proc;

Column[LinearRecurrence[{4,-5},{2,6},40]] (* Vincenzo Librandi, Jan 15 2012 *)

Vec(2*(1-x)/(1-4*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 15 2012

G.f.: 2*(1-x)/(1-4*x+5*x^2). [Colin Barker, Jan 14 2012]

cp's OEIS Frontend

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A121625 Real part of (n + n*i)^n.

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A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

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A193410 Expansion of (1-3*x)/(1-6*x+18*x^2).

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A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

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A077860 Expansion of 1/((1 - 2x + 2x^2)*(1-x)).

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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.

A102486 a(n) = 4a(n-1) - 5a(n-2).