A146559 -id:A146559 - cp's OEIS frontend

A099087 Expansion of 1/(1 - 2x + 2x^2).

1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216

Offset: 0

Paul Barry, Sep 24 2004

Yet another variation on A009545.
Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(Pi*n/2). See A009116.
Binomial transform of A057077. - R. J. Mathar, Nov 04 2008
Partial sums of A146559. - Philippe Deléham, Dec 01 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
Also the inverse Catalan transform of A000079. - Arkadiusz Wesolowski, Oct 26 2012

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Karl Dilcher and Maciej Ulas, Divisibility and Arithmetic Properties of a Class of Sparse Polynomials, arXiv:2008.13475 [math.NT], 2020. See Table 1, 1st column, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-2).

Cf. A009545, A146559.

a:=[1,2];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Mar 16 2019

I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1) - Self(n-2)): n in [1..50]]; // G. C. Greubel, Mar 16 2019

CoefficientList[Series[1/(1 -2x +2x^2), {x, 0, 50}], x] (* Michael De Vlieger, Dec 24 2015 *)

x='x+O('x^50); Vec(1/(1-2*x+2*x^2)) \\ Altug Alkan, Dec 24 2015

[lucas_number1(n,2,2) for n in range(1, 50)] # Zerinvary Lajos, Apr 23 2009

E.g.f.: exp(x)*(cos(x) + sin(x)).
a(n) = 2^(n/2)*(cos(Pi*n/4) + sin(Pi*n/4)).
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, k-i)*binomial(n, i) *(-1)^(k-i).
a(n) = 2*(a(n-1) - a(n-2)).
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = (1-i)^(n-1) + (1+i)^(n-1) where i=sqrt(-1).
a(n) = 2 Sum_{k=0..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. (End)
a(n) = Sum_{k=0..n} A109466(n,k)*2^k. - Philippe Deléham, Oct 28 2008
E.g.f.: (cos(x)+sin(x))*exp(x) = G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - Philippe Deléham, Feb 13 2013
a(n) = Sum_{j=0..n} binomial(n, j)*(-1)^binomial(j, 2); this is the case m=2 and z=-1 of f(m,n)(z) = Sum_{j=0..n} binomial(n, j)*z^binomial(j, m). See Dilcher and Ulas. - Michel Marcus, Sep 01 2020

Signs added by N. J. A. Sloane, Nov 14 2006

A126930 Inverse binomial transform of A005043.

Original entry on oeis.org

1, -1, 2, -3, 6, -10, 20, -35, 70, -126, 252, -462, 924, -1716, 3432, -6435, 12870, -24310, 48620, -92378, 184756, -352716, 705432, -1352078, 2704156, -5200300, 10400600, -20058300, 40116600, -77558760, 155117520, -300540195, 601080390, -1166803110

Offset: 0

Philippe Deléham, Mar 17 2007

Successive binomial transforms are A005043, A000108, A007317, A064613, A104455. Hankel transform is A000012.
Moment sequence of the trace of the square of a random matrix in USp(2)=SU(2). If X=tr(A^2) is a random variable (a distributed with Haar measure) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008
From Tom Copeland, Nov 08 2014: (Start)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Mobius) transformation P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). The Motzkin sums, or Riordan numbers, A005043 are generated by Mot(x)=C[P(x,1)]. One could, of course, choose the Riordan numbers as the parent sequence.
O.g.f.: G(x) = C[P[P(x,1),1]1] = C[P(x,2)] = (1-sqrt(1-4*x/(1+2*x)))/2 = x - x^2 + 2 x^3 - ... = Mot[P(x,1)].
Ginv(x) =  Pinv[Cinv(x),2] = P[Cinv(x),-2] = x(1-x)/[1-2x(1-x)] = (x-x^2)/[1-2(x-x^2)] = x*A146559(x).
Cf. A091867 and A210736 for an unsigned version with a leading 1. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 100.

Cf. A126120, A126869.

Cf. A000108, A005043, A146559, A091867, A210736.

egf := BesselI(0,2*x) - BesselI(1,2*x):
seq(n!*coeff(series(egf,x,34),x,n),n=0..33); # Peter Luschny, Dec 17 2014

CoefficientList[Series[(1 + 2 x - Sqrt[1 - 4 x^2])/(2 x (1 + 2 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)
Table[2^n Hypergeometric2F1[3/2, -n, 2, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 02 2015 *)

x='x+O('x^50); Vec((1+2*x-sqrt(1-4*x^2))/(2*x*(1+2*x))) \\ Altug Alkan, Nov 03 2015

a(n) = (-1)^n*C(n, floor(n/2)) = (-1)^n*A001405(n).
a(2*n) = A000984(n), a(2*n+1) = -A001700(n).
a(n) = (1/Pi)*Integral_{t=0..Pi}(2cos(2t))^n*2sin^2(t) dt. - Andrew V. Sutherland, Feb 29 2008, Mar 09 2008
a(n) = (-2)^n + Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. - Philippe Deléham, Dec 12 2009
G.f.: (1+2*x-sqrt(1-4*x^2))/(2*x*(1+2*x)). - Philippe Deléham, Mar 01 2013
O.g.f.: (1 + x*c(x^2))/(1 + 2*x), with the o.g.f. c(x) for the Catalan numbers A000108. From the o.g.f. of the Riordan type Catalan triangle A053121. This is the rewritten g.f. given in the previous formula. This is G(-x) with the o.g.f. G(x) of A001405. - Wolfdieter Lang, Sep 22 2013
D-finite with recurrence (n+1)*a(n) +2*a(n-1) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Dec 04 2013
Recurrence (an alternative): (n+1)*a(n) = 8*(n-2)*a(n-3) + 4*(n-2)*a(n-2) + 2*(-n-1)*a(n-1), n>=3. - Fung Lam, Mar 22 2014
Asymptotics: a(n) ~ (-1)^n*2^(n+1/2)/sqrt(n*Pi). - Fung Lam, Mar 22 2014
E.g.f.: BesselI(0,2*x) - BesselI(1,2*x). - Peter Luschny, Dec 17 2014
a(n) = 2^n*hypergeom([3/2,-n], [2], 2). - Vladimir Reshetnikov, Nov 02 2015
G.f. A(x) satisfies: A(x) = 1/(1 + 2*x) + x*A(x)^2. - Ilya Gutkovskiy, Jul 10 2020

A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -2, 0, 1, 0, -2, -3, 0, 1, 1, 1, -3, -4, 0, 1, 1, 4, 3, -4, -5, 0, 1, 0, 3, 9, 6, -5, -6, 0, 1, -1, -2, 5, 16, 10, -6, -7, 0, 1, -1, -6, -9, 6, 25, 15, -7, -8, 0, 1, 0, -4, -18, -24, 5, 36, 21, -8, -9, 0, 1, 1, 3, -7, -39, -50, 1, 49, 28, -9, -10, 0, 1, 1, 8

Offset: 0

Ralf Stephan, Mar 17 2005

Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - Paul Barry, Jun 21 2008

			1
0,1
-1,0,1
-1,-2,0,1
0,-2,-3,0,1
1,1,-3,-4,0,1
1,4,3,-4,-5,0,1
0,3,9,6,-5,-6,0,1
-1,-2,5,16,10,-6,-7,0,1
-1,-6,-9,6,25,15,-7,-8,0,1

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

Row sums are A009116 with different signs.
Row sums are A146559(n).
Cf. A005043, A091867, A030528, A000108.

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

T(n,m):=sum(binomial(m,j)*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1),k,0,n)*(-1)^(m-j),j,0,m); /* Vladimir Kruchinin, Apr 08 2011 */

T(n,m) = sum(j=0..m, binomial(m,j)*sum(k=0..n, binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1))*(-1)^(m-j)). - Vladimir Kruchinin, Apr 08 2011
T(n,m) = sum(k=ceiling((n-m-1)/2)..n-m, binomial(k+m,m)*binomial(k+1,n-k-m)*(-1)^(n-k-m)). - Vladimir Kruchinin, Dec 17 2011
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 20 2013
T(n+5,n) = (n+1)^2. - Philippe Deléham, Feb 20 2013
From Tom Copeland, Nov 04 2014: (Start)
O.g.f.: G(x,t) = Pinv[Cinv(x),t+1] = Cinv(x) / [1 - (t+1)Cinv(x)] = x*(1-x) / [1-(t+1)x(1-x)] = x + t * x^2 + (-1 + t^2) * x^3 + ..., where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108 and Pinv(x,t) = -P(-x,t) = x/(1-t*x) is the inverse of P(x,t) = x/(1+x*t).
Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2.
The inverse in x of G(x,t) with t replaced by -t is the o.g.f. of A091867, and G(x,t-1) is a signed version of the (mirrored) Fibonacci polynomials A030528. (End)

A210736 Expansion of (1 + sqrt( (1 + 2x) / (1 - 2x))) / 2 in powers of x.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432, 1352078, 2704156, 5200300, 10400600, 20058300, 40116600, 77558760, 155117520, 300540195, 601080390, 1166803110, 2333606220, 4537567650

Offset: 0

Michael Somos, May 10 2012

nonn

Hankel transform is period 4 sequence [ 1, 0, -1, 0, ...] A056594 and the Hankel transform of sequence omitting a(0) is the all 1s sequence A000012. This is the unique sequence with that property.
Series reversion of x*A(x) apparently yields x*A036765(-x). - R. J. Mathar, Sep 24 2012
a(n) is the number of length n words on {-1,1} such that the sum of any of its prefixes is always positive.  Cf. A001405 where the sum of all prefixes is nonnegative. - Geoffrey Critzer, Jul 08 2013

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 20*x^7 + 35*x^8 + 70*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 25.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; page 77.

Essentially the same as A001405.
Cf. A056594, A138350, A210628, A211278.
Cf. A000108, A091867, A146559, A126930.

nn=36; d=(1-(1-4x^2)^(1/2))/(2x^2);CoefficientList[Series[1/(1-x d),{x,0,nn}],x] (* Geoffrey Critzer, Jul 08 2013 *)
CoefficientList[Series[2 x / (-1 + 2 x + Sqrt[1 - 4 x^2]), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

{a(n) = if( n<1, n==0, binomial( n - 1, (n - 1)\2))};

{a(n) = polcoeff( (1 + sqrt( (1 + 2*x) / (1 - 2*x) + x * O(x^n))) / 2, n)};

G.f.: 2 * x / (-1 + 2*x + sqrt(1 - 4*x^2)).
G.f. A(x) satisfies A(x) = A(x)^2 - x / (1 - 2*x).
G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 / (1 - x).
G.f. A(x) satisfies A(1/3) = (1 + sqrt(5))/2.
G.f. A(x) = 1 + x / (1 - 2*x + x / A(x)).
G.f. A(x) = 1 + x / (1 - x / (1 - x / (1 + x / A(x)))).
G.f. A(x) = 1 + x * A001405(x). a(n+1) = A001405(n).
Convolution inverse is A210628. Partial sums is A072100.
Binomial transform with offset 1 is A211278 with offset 1. a(n+2) * a(n) - a(n+1)^2 = A138350(n-1).
a(n) = (-1)^floor(n/2)*hypergeom2F1([1-n, -n],[1],-1). - Peter Luschny, Sep 01 2012
D-finite with recurrence: n*a(n) -2*a(n-1) +4*(2-n)*a(n-2)=0. - R. J. Mathar, Sep 14 2012
G.f. A(x) = 1 / (1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...)))). - Michael Somos, Jan 02 2013
G.f.: 1/(1 - x*C(x)) where C(x) is the o.g.f. for A126120. - Geoffrey Critzer, Jul 08 2013
a(n) ~ 2^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 01 2014
G.f.: A(x) = 1 - x/(- 1 + x/A(-x)). - Arkadiusz Wesolowski, Feb 28 2014
From Tom Copeland, Nov 07 2014: (Start)
Setting a(0)=0 here, we have a signed version in A126930 and
O.g.f. G(x)=[-1+sqrt(1+4*x/(1-2x))]/2 = x + x^2 + 2 x^3 + ... = -C[-P(P(x,-1),-1)]= -C[-P(x,-2)] where C(x)= [1-sqrt(1-4*x)]/2= x + x^2 + 2 x^3 + ... = A000108(x) with inverse Cinv(x)=x*(1-x), and P(x,t)= x/(1 + t*x) with inverse P(x,-t).
These types of arrays are from linear fractional transformations of C(x). See A091867.
Ginv(x) = P[-Cinv(-x),2] = x*(1+x)/(1+2*x*(1-x))= (x+x^2)/(1+2(x+x^2)) (see A146559). (End)

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A307039 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, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -2, 0, 1, 1, 1, 0, -4, 0, 1, 1, 1, 1, -3, -4, 0, 1, 1, 1, 1, 0, -9, 0, 0, 1, 1, 1, 1, 1, -4, -18, 8, 0, 1, 1, 1, 1, 1, 0, -14, -27, 16, 0, 1, 1, 1, 1, 1, 1, -5, -34, -27, 16, 0, 1, 1, 1, 1, 1, 1, 0, -20, -68, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, -6, -55, -116, 81, -32, 0

Offset: 0

Seiichi Manyama, Mar 21 2019

			Square array begins:
   1,  1,   1,    1,    1,   1,   1,  1, ...
   0,  1,   1,    1,    1,   1,   1,  1, ...
   0,  0,   1,    1,    1,   1,   1,  1, ...
   0, -2,   0,    1,    1,   1,   1,  1, ...
   0, -4,  -3,    0,    1,   1,   1,  1, ...
   0,  0, -18,  -14,   -5,   0,   1,  1, ...
   0,  8, -27,  -34,  -20,  -6,   0,  1, ...
   0, 16, -27,  -68,  -55, -27,  -7,  0, ...
   0, 16,   0, -116, -125, -83, -35, -8, ...

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

Columns 1-9 give A000007, A146559, A057681, A099586, A289306, A307040, A307041, A307044, A307045.

Cf. A306846, A306914.

T[n_, k_] := Sum[(-1)^j * Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 13}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n,k*j).

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882

Offset: 0

Gary W. Adamson, Apr 05 2008

Imaginary part of (2 + i)^n gives A099456.
Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.
Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).

Seiichi Manyama, Table of n, a(n) for n = 0..2862 (first 201 terms from Vincenzo Librandi)
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-5).

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

[ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011

restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009

Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)

a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */

Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

[lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Real part of (2 + i)^n, i^2 = -1.
Term (1,1) of matrix [2,-1; 1,2]^n.
(a(n))^2 + (A099456(n))^2 = 5^n.
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: (1-2x) /(1-4x+5x^2).
a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)
E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009
a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010
a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008
2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011
a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009
Added a(0)=1 by Michael Somos, Dec 26 2010
Edited by Franklin T. Adams-Watters, Apr 10 2011

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A213421 Real part of Q^n, Q being the quaternion 2+i+j+k.

Original entry on oeis.org

1, 2, 1, -10, -47, -118, -143, 254, 2017, 6290, 11041, 134, -76751, -307942, -694511, -622450, 2371777, 13844258, 38774593, 58188566, -38667887, -561991510, -1977290831, -3975222754, -2059855199, 19587138482

Offset: 0

Stanislav Sykora, Jun 11 2012

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras Vol. 29, No. 3 (2019), Article 54.
Wikipedia, Lucas sequence

Cf. A168175, A087455, A088138, A128018, A146559.

#A213421
seq(simplify(1/2*((2+I*sqrt(3))^n+(2-I*sqrt(3))^n)), n = 0 .. 25); # Peter Bala, Mar 29 2015

QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);}
Q=QuaternionToN(2,1,1,1,1000);
for(n=1,#Q[,1],write("A213421.txt",n-1," ",Q[n,1]));

Conjecture: G.f. (1-2x)/(1-4x+7x^2). a(n) = A168175(n)-2*A168175(n-1). - R. J. Mathar, Jun 25 2012
From Peter Bala, Mar 29 2015: (Start)
The above o.g.f. is correct; this is the Lucas sequence V_n(4,7).
a(n) = Re( (2 + sqrt(3)*i)^n )= 1/2*( (2 + sqrt(3)*i)^n + (2 - sqrt(3)*i)^n ).
a(n) = 1/2 * trace( [ 2 + i, 1 + i; -1 + i, 2 - i ]^n ) = 1/2 * trace( [ 2 , sqrt(3)*i ; sqrt(3)*i, 2 ]^n ).
a(n) = 4*a(n-1) - 7*a(n-2) with a(0) = 1, a(1) = 2. (End)

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

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A099087 Expansion of 1/(1 - 2*x + 2*x^2).

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A126930 Inverse binomial transform of A005043.

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A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

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A210736 Expansion of (1 + sqrt( (1 + 2*x) / (1 - 2*x))) / 2 in powers of x.

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A124182 A skewed version of triangular array A081277.

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A307039 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).

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A139011 Real part of (2 + i)^n, where i = sqrt(-1).

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A099087 Expansion of 1/(1 - 2x + 2x^2).

A210736 Expansion of (1 + sqrt( (1 + 2x) / (1 - 2x))) / 2 in powers of x.

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).