A052542 -id:A052542 - cp's OEIS frontend

A097564 a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.

0, 1, 1, 2, 1, 3, 4, 3, 7, 10, 7, 17, 24, 17, 41, 58, 41, 99, 140, 99, 239, 338, 239, 577, 816, 577, 1393, 1970, 1393, 3363, 4756, 3363, 8119, 11482, 8119, 19601, 27720, 19601, 47321, 66922, 47321, 114243, 161564, 114243, 275807, 390050, 275807, 665857

Offset: 0

Gerald McGarvey, Aug 27 2004

The sequences a(2), a(5), ... a(1+3*n) ... and a(4), a(7), ... a(4 + 3n) ... are both A001333 (numerators of continued fraction convergents to sqrt(2)). The sequence a(0), a(3), a(6), ... a(3+3*n) ... is twice A000129 (the Pell nos. or the denominators of continued fraction convergents to sqrt(2).), also is A052542 starting w/ offset 1.

Colin Barker, Table of n, a(n) for n = 0..1000
D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1).

[n le 2 select n-1 else (Self(n-1) mod 2)*Self(n-1)+Self(n-2): n in [1..50]]; // Bruno Berselli, Jun 02 2016

m:=50; S:=series( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 20 2021

nxt[{a_,b_}]:={b,Mod[b,2]*b+a}; NestList[nxt,{0,1},50][[All,1]] (* or *) LinearRecurrence[{0,0,2,0,0,1},{0,1,1,2,1,3},50] (* Harvey P. Dale, Aug 15 2017 *)

concat(0, Vec(x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6) + O(x^100))) \\ Colin Barker, Jun 02 2016

def A097564_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6) ).list()
A097564_list(50) # G. C. Greubel, Apr 20 2021

From Colin Barker, Jun 01 2016: (Start)
a(n) = 2*a(n-3) + a(n-6) for n>5.
G.f.: x*(1+x+2*x^2-x^3+x^4) / (1-2*x^3-x^6). (End)

A100434 Expansion of g.f. (1+x)(3+x)/(1+6x^2+x^4).

Original entry on oeis.org

3, 4, -17, -24, 99, 140, -577, -816, 3363, 4756, -19601, -27720, 114243, 161564, -665857, -941664, 3880899, 5488420, -22619537, -31988856, 131836323, 186444716, -768398401, -1086679440, 4478554083, 6333631924, -26102926097, -36915112104, 152139002499, 215157040700

Offset: 0

N. J. A. Sloane, Nov 21 2004, suggested by correspondence from Creighton Dement

From Creighton Dement, Dec 18 2004: (Start)
Define the following sequences:
b(2n) = c(2n+1), b(2n+1) = c(2n); (c(n)) = (1, -3, -7, 17, 41, -99, -239, 577, 1393, -3363, -8119, 19601, 47321). This is the sequence A001333, apart from signs. Then c(2n) = ((-1)^n)*A002315(n) and c(2n+1) = ((-1)^(n+1))*A001541(n+1).
(d(n)) = (2, 4, -10, -24, 58, 140, -338, -816, 1970, 4756, -11482, -27720). This is A052542, apart from signs. Also, d(2n) = ((-1)^n)*A075870(n), d(2n+1) = ((-1)^n)*A005319(n+1).
(e(n)) = (1, -1, -5, 5, 29, -29, -169, 169, 985, -985, -5741, 5741, 33461, -33461), e(2n) = d(2n)/2, e(2n+1) = - d(2n)/2.
(f(n)) = (2, 2, -12, -12, 70, 70, -408, -408, 2378, 2378, -13860, -13860, ) f(2n) = f(2n+1) = d(2n+1)/2.
(g(n)) = (0, -3, 0, 17, 0, -99, 0, 577, 0, -3363, 0, 19601, 0, -114243, 0, 665857), g(2n) = 0, g(2n+1) = c(2n+1).
Then a(2n) = - c(2n+1), a(2n+1) = d(2n+1) and we have the following conjectures: c(n) + d(n) = e(n) + f(n) = g(n) + a(n); c(n) + d(n) = b(n). In other words, the sequences (c(n) + d(n)) = (e(n) + f(n)) = (g(n) + h(n)) all represent the sequence c with even- and odd-indexed terms reversed. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-6,0,-1).

Bisections give A001541, A005319.

Cf. A000034, A001333, A002315, A052542, A075870, A126354.

I:=[3,4,-17,-24]; [n le 4 select I[n] else -6*Self(n-2)-Self(n-4): n in [1..40]]; // G. C. Greubel, Apr 09 2023

LinearRecurrence[{0,-6,0,-1}, {3,4,-17,-24}, 41] (* G. C. Greubel, Apr 09 2023 *)

@CachedFunction
def a(n): # a = A100434
    if (n<4): return (3,4,-17,-24)[n]
    else: return -6*a(n-2) - a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Apr 09 2023

a(n) = (-1)^floor(n/2)*A000034(n)*A126354(n+3). - R. J. Mathar, Mar 08 2009

a(n) = -2*a(n-1) - 3*a(n-2) if n is even; a(n) = (4*a(n-1) - a(n-2))/3 if n is odd. - R. J. Mathar, Jun 18 2014

A136421 a(n) = floor((x^n - (1-x)^n)/sqrt(2)+ 1/2) where x = (sqrt(2)+1)/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 21, 25, 31, 37, 44, 54, 65, 78, 94, 114, 138, 166, 200, 242, 292, 352, 425, 514, 620, 748, 903, 1090, 1316, 1589, 1918, 2315, 2794, 3373, 4072, 4915, 5933, 7162, 8645, 10436, 12597, 15206, 18355, 22156, 26745

Offset: 1

Cino Hilliard, Apr 01 2008

nonn

This is analogous to the closed form of the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x=(sqrt(2)+1)/2, a(n)/a(n-1) -> x.

G. C. Greubel, Table of n, a(n) for n = 1..10000

[Floor(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2^n*Sqrt(2))+ 1/2): n in [2..50]]; // G. C. Greubel, Oct 02 2018

Table[Floor[Fibonacci[n, 2]/2^(n-1) +1/2], {n,1,50}] (* G. C. Greubel, Oct 02 2018 *)

fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5);
g(n,r) = for(m=1,n,print1(fib(m,r)", "));
g(30,2)

The general form of x is (sqrt(r)+1)/2, r=1,2,3...

a(n) = floor(b(n)/2^n) where b(n) = A052542(n) + 2^(n-1) = 4*b(n-1) - 3*b(n-2) - 2*b(n-3). - R. J. Mathar, Sep 10 2016

A052622 E.g.f. (1-x^2)/(1-2x-x^2).

Original entry on oeis.org

1, 2, 8, 60, 576, 6960, 100800, 1703520, 32901120, 714873600, 17258572800, 458324697600, 13277924352000, 416724685977600, 14084873439436800, 510058387238400000, 19702238017093632000, 808611973910028288000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 568

spec := [S,{S=Sequence(Prod(Union(Z,Z),Sequence(Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-x^2)/(1-2x-x^2),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Mar 04 2013 *)

E.g.f.: (-1+x^2)/(-1+2*x+x^2)
Recurrence: {a(0)=1, a(1)=2, a(2)=8, (-2-n^2-3*n)*a(n) +(-4-2*n)*a(n+1) +a(n+2)=0}
Sum(-1/2*(-1+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z+_Z^2))*n!
a(n) = n!*((1+sqrt(2))^n - (1-sqrt(2))^n)/sqrt(2). - Vaclav Kotesovec, Oct 05 2013
a(n)=n!*A052542(n). - R. J. Mathar, Jun 03 2022

A062507 Table by antidiagonals related to partial sums and differences of Pell numbers (A000129).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 10, 7, 5, 3, 1, 0, 24, 17, 12, 8, 4, 1, 0, 58, 41, 29, 20, 12, 5, 1, 0, 140, 99, 70, 49, 32, 17, 6, 1, 0, 338, 239, 169, 119, 81, 49, 23, 7, 1, 0, 816, 577, 408, 288, 200, 130, 72, 30, 8, 1, 0, 1970, 1393, 985, 696, 488, 330, 202, 102

Offset: 0

Henry Bottomley, Jul 09 2001

			Rows start (0,1,0,2,4,10,...), (0,1,1,3,7,17,...), (0,1,2,5,12,29,...) etc.

Rows are effectively A052542, A001333, A000129, A048739, A048776. Columns are effectively A000004, A000012, A001477, A022856.

T(n, k) =T(n, k-1)+T(n-1, k) =2T(n, k-1)+T(n, k-2)+C(n+k-3, k) for n>2.

A068515 A measure of how close the square root of 2 is to rational numbers.

Original entry on oeis.org

2, -12, 12, -12, 70, 12, -70, 26, -33, 70, -25, -408, 34, -70, 70, -43, 408, 39, -146, 70, -70, 195, -49, -408, 70, -113, 147, -70, 2378, 70, -195, 126, -100, 408, 70, -408, 114, -146, 253, -93, -2378, 106, -228, 195, -125, 855, 100, -408, 165, -173, 408, -113, -1135, 147, -252, 286, -146, 2378, 135, -408

Offset: 1

Henry Bottomley, Mar 19 2002

sign

New peaks (in absolute terms) occur when n is a Pell number (1,2,5,12,29,70,... A000129) and take alternate Pell values with alternating signs (2,-12,70,-408,2378,-13860,... A001542). Each new peak (after the first) appears twice (with different signs) before the next peak, when n is a numerator of a continued fraction convergent to sqrt(2) (3,7,17,41,99,... A001333) and when n is twice a Pell number (4,10,24,58,140,... A052542).

			a(5) = round[1/(sqrt(2)-round[sqrt(2)*5]/5)] = round[1/(sqrt(2)-7/5)] = round[70.355] = 70, i.e. sqrt(2) is about 1/70 more than the nearest multiple of 1/5.

Cf. A066212.

a(n) =round[1/(sqrt(2)-round[sqrt(2)*n]/n)] =round[1/(sqrt(2)-A022846(n)/n)] where sqrt(2)=1.41421356...

A346631 Number of strongly asymmetric Boolean nested canalizing functions with n variables.

Original entry on oeis.org

4, 24, 240, 2880, 41760

Offset: 2

Yuan Li, Jul 25 2021

nonn

Yuan Li, Frank Ingram, Huaming Zhang, Certificate complexity and symmetric properties of nested canalizing functions, arXiv:2001.09094 [cs.DM], 2020-2021

Comments from the Editors: It appears that this is essentially n!*A163271(n) (see also A052542), and also essentially 2*n!*A000129(n) (see also A215928). It also appears to match 2*A052580.

a(n) = n!((1+sqrt(2))^{n-1}-(1-sqrt(2))^{n-1})/sqrt(2).

cp's OEIS Frontend

A097564 a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.

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A100434 Expansion of g.f. (1+x)*(3+x)/(1+6*x^2+x^4).

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A136421 a(n) = floor((x^n - (1-x)^n)/sqrt(2)+ 1/2) where x = (sqrt(2)+1)/2.

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A052622 E.g.f. (1-x^2)/(1-2x-x^2).

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Maple

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A062507 Table by antidiagonals related to partial sums and differences of Pell numbers (A000129).

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A068515 A measure of how close the square root of 2 is to rational numbers.

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A346631 Number of strongly asymmetric Boolean nested canalizing functions with n variables.

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A100434 Expansion of g.f. (1+x)(3+x)/(1+6x^2+x^4).