A072100 -id:A072100 - cp's OEIS frontend

A099323 Expansion of (sqrt(1+3x) + sqrt(1-x))/(2sqrt(1-x)).

1, 1, 0, 1, -1, 3, -6, 15, -36, 91, -232, 603, -1585, 4213, -11298, 30537, -83097, 227475, -625992, 1730787, -4805595, 13393689, -37458330, 105089229, -295673994, 834086421, -2358641376, 6684761125, -18985057351, 54022715451, -154000562758, 439742222071, -1257643249140

Offset: 0

Paul Barry, Oct 12 2004

Binomial transform is A072100.
Signed Motzkin numbers with an additional leading 1.
Inverse binomial transform of A001405 gives this without the initial 1. So does the binomial transform of (-1)^n*A000108(n) = [1,-1,2,-5,14,-42,...]. - Philippe Deléham, Mar 20 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC'02 Melbourne, 2002.

Cf. A000108, A001405, A005043, A072100.

A099323:= func< n | (&+[(-1)^k*Binomial(n-1, k)*Catalan(k): k in [0..n]]) >;
[A099323(n): n in [0..40]]; // G. C. Greubel, Nov 25 2021

with(PolynomialTools): CoefficientList(convert(taylor((sqrt(1 + 3*x) + sqrt(1 - x))/2/sqrt(1 - x), x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]),{x,0,40}],x] (* Harvey P. Dale, Feb 06 2015 *)

[sum((-1)^k*binomial(n-1, k)*catalan_number(k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Nov 25 2021

a(n) = 0^n + Sum_{k=0..n-1} binomial(n-1,k)*(-1)^k*C(k), where C(k) is the k-th Catalan number.
G.f.: 1 + x/(1-sqrt(x))/G(0), where G(k)= 1 +  sqrt(x)/(1  -  sqrt(x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 28 2013
D-finite with recurrence: n*a(n) + 2*(n-2)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - R. J. Mathar, Oct 10 2014
a(n) ~ -(-1)^n * 3^(n + 1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 31 2017

Edited by N. J. A. Sloane, Oct 05 2009

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)

A004074 a(n) = 2*A004001(n) - n, where A004001 is the Hofstadter-Conway $10000 sequence.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 7, 8, 7, 8, 7, 6, 7, 6, 5, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 11, 12, 11, 10, 11, 12, 13, 12, 13, 14, 13, 14, 13, 12

Offset: 1

N. J. A. Sloane

nonn

The sequence is 0 at 2^n for n = 1, 2, 3, ... The maximum value between 2^n and 2^(n+1) appears to be A072100(n). - T. D. Noe, Jun 04 2012

Hofstadter shows the plot of sequence A004001(n)-(n/2) at point 10:52 of the part two of DIMACS lecture. This sequence is obtained by doubling those values, thus producing only integers. Cf. also A249071. - Antti Karttunen, Oct 22 2014

T. D. Noe (first 10000 terms) and Antti Karttunen, Table of n, a(n) for n = 1..16384
D. R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, October 10 2014; Part 1, Part 2.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Wikipedia, Blancmange curve
Index entries for Hofstadter-type sequences

Cf. A004001, A082590, A213057.

Cf. also A249071 (gives the even bisection halved), A233270 (also has a similar Blancmange curve appearance).

Clear[a]; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[2*a[n] - n, {n, 100}] (* T. D. Noe, Jun 04 2012 *)

(define (A004074 n) (- (* 2 (A004001 n)) n)) ;; Other code as in A004001. - Antti Karttunen, Oct 22 2014

a(2^n)=0; for n >= 1, Sum_{i=2^(n-1)..2^n} a(i) = A082590(n-2). - Benoit Cloitre, Jun 04 2004

More terms from Benoit Cloitre, Jun 04 2004

A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

Original entry on oeis.org

1, 0, -1, 3, -8, 22, -64, 196, -625, 2055, -6917, 23713, -82499, 290511, -1033411, 3707851, -13402696, 48760366, -178405156, 656043856, -2423307046, 8987427466, -33453694486, 124936258126, -467995871776, 1757900019100, -6619846420552, 24987199492704, -94520750408708

Offset: 0

Paul Barry, Oct 12 2004

Binomial transform is A099323. Second binomial transform is A072100.

Hankel transform is A049347. - Paul Barry, Aug 10 2009

Robert Israel, Table of n, a(n) for n = 0..1668
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Cf. A014138.

f:= gfun:-rectoproc({(2+4*n)*a(n)+(4+5*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 0}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 27 2018

CoefficientList[Series[(1+Sqrt[1+4x])/(2(1+x)),{x,0,40}],x] (* Harvey P. Dale, Jan 30 2014 *)

a(n) = Sum_{k=0..2n} (2*0^(2n-k)-1)*C(k,floor(k/2)). - Paul Barry, Aug 10 2009
|a(n+2)| = A091491(n+2,2). - Philippe Deléham, Nov 25 2009
G.f.: T(0)/(2+2*x), where T(k) = k+2 - 2*x*(2*k+1) + 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
D-finite with recurrence: (2+4*n)*a(n) + (4+5*n)*a(n+1) + (n+2)*a(n+2) = 0. - Robert Israel, Mar 27 2018

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A099323 Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).

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

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A004074 a(n) = 2*A004001(n) - n, where A004001 is the Hofstadter-Conway $10000 sequence.

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A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

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A099323 Expansion of (sqrt(1+3x) + sqrt(1-x))/(2sqrt(1-x)).

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