A001079 -id:A001079 - cp's OEIS frontend

A165293 Inverse of A038303, and generalization of A130595.

1, 10, -1, 100, -20, 1, 1000, -300, 30, -1, 10000, -4000, 600, -40, 1, 100000, -50000, 10000, -1000, 50, -1, 1000000, -600000, 150000, -20000, 1500, -60, 1, 10000000, -7000000, 2100000, -350000, 35000, -2100, 70

Offset: 1

Mark Dols, Sep 13 2009

Rows sum up to A001019 (powers of 9), diagonals to A004189, a generalization of A010892 (the inverse Fibonacci). Ratio of diagonal sums converges to a decimal sequence: A000108 (Catalan numbers), which is the squared difference of sqrt(2) and sqrt(3), or 5-sqrt(24). Ratio between first binomial transform (A054320 and A138288)of A004189, converges to sqrt(2/3). 1/(2*sqrt(24)) gives A000984 (central binomial coefficients) as a decimal sequence.

Triangle T(n,k), read by rows, given by [10,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			Triangle begins:
      1;
     10,    -1;
    100,   -20,   1;
   1000,  -300,  30,  -1;
  10000, -4000, 600, -40, 1;

Cf. A007318, A130595, A038303, A004189, A010892, A001079, A054320, A138288, A041041, A000108.

Sum_{k=0..n} T(n,k)*x^k = (10-x)^n. - Philippe Deléham, Dec 15 2009

G.f.: x*y/(1-10*x+x*y). - R. J. Mathar, Aug 11 2015

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Original entry on oeis.org

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030

Offset: 0

Gary Detlefs, Feb 20 2012

Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2.
.
   k             x                        y
   -  -----------------------  -----------------------
A001542 {0,2,6,-1}       A001541 {1,3,6,-1}
A001353 {0,1,4,-1}       A001075 {1,2,4,-1}
A060645 {0,4,18,-1}      A023039 {1,9,18,-1}
A001078 {0,2,10,-1}      A001079 {1,5,10,-1}
A001080 {0,3,16,-1}      A001081 {1,8,16,-1}
A001109 {0,1,6,-1}       A001541 {1,3,6,-1}
A084070 {0,1,38,-1}      A078986 {1,19,38,-1}
A001084 {0,3,20,-1}      A001085 {1,10,20,-1}
A011944 {0,2,14,-1}      A011943 {1,7,14,-1}
A075871 {0,180,1298,-1}  A114047 {1,649,1298,-1}
A068204 {0,4,30,-1}      A069203 {1,15,30,-1}
A001090 {0,1,8,-1}       A001091 {1,4,8,-1}
A121740 {0,8,66,-1}      A099370 {1,33,66,-1}
A202299 {0,4,34,-1}      A056771 {1,17,34,-1}
A174765 {0,39,340,-1}    A114048 {1,179,340,-1}
a(n)    {0,2,18,-1}      A023039 {1,9,18,-1}
A174745 {0,12,110,-1}    A114049 {1,55,110,-1}
A174766 {0,42,394,-1}    A114050 {1,197,394,-1}
A174767 {0,5,48,-1}      A114051 {1,24,48,-1}
A004189 {0,1,10,-1}      A001079 {1,5,10,-1}
A174768 {0,10,102,-1}    A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

Bruno Berselli, Table of n, a(n) for n = 0..500
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A023039, A049660, A079962.

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019

readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;

LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)

makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

a(n) = 18*a(n-1) - a(n-2).
From Bruno Berselli, Feb 21 2012: (Start)
G.f.: 2*x/(1-18*x+x^2).
a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019
For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021

A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.

Original entry on oeis.org

0, 0, 8, 48, 160, 400, 840, 1568, 2688, 4320, 6600, 9680, 13728, 18928, 25480, 33600, 43520, 55488, 69768, 86640, 106400, 129360, 155848, 186208, 220800, 260000, 304200, 353808, 409248, 470960, 539400, 615040, 698368, 789888, 890120, 999600

Offset: 0

Henry Bottomley, Jul 14 2002

			a(2)=8 since for a random walk with absorbing boundaries at +2 or -2, the probability of first reaching a boundary at time t=2 is 1/2, at t=4 is 1/4, at t=6 is 1/8, at t=8 is 1/16, etc., giving a mean of 2/2 + 4/4 + 6/8 + 8/16 + ... = 4 and a variance of 2^2/2 + 4^2/4 + 6^2/8 + 8^2/16 + ... - 4^2 = 24 - 16 = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290 (i.e., n^2) for mean time. A072818(n)=sqrt(a(A001079(n))) attempts to identify the integer standard deviations.

[n^2*(n^2-1)*2/3: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

CoefficientList[Series[8 (1 + x) x^2/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, Jul 02 2019 *)

a(n) = n^2*(n^2 - 1)*2/3 = 4*A008911(n) = 8*A002415(n) = A069971(n, n).
G.f.: 8*(1 + x)*x^2/(1 - x)^5. - Arkadiusz Wesolowski, Feb 08 2012
E.g.f.: 2*exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 12 2021
a(n) = 2*n * A007290(n+1). - C.S. Elder, Jan 09 2024

A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 60, 594, 5880, 58206, 576180, 5703594, 56459760, 558894006, 5532480300, 54765908994, 542126609640, 5366500187406, 53122875264420, 525862252456794, 5205499649303520, 51529134240578406, 510085842756480540, 5049329293324226994, 49983207090485789400

Offset: 0

N. J. A. Sloane, Sep 21 2006

Kekulé numbers for the benzenoids P''(n).
a(n) are the integer square roots of A032528(m) - 1. A001079 gives the value of m where these roots occur. Also see A122652. - Richard R. Forberg, Aug 05 2013
Numbers n such that 6*n^2 + 9 is a square. - Colin Barker, Mar 17 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 301).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,-1).

CoefficientList[Series[(6 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1},{0,6},30] (* Harvey P. Dale, Dec 16 2014 *)

a(n)=if(n<2,(n%2)*6,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

G.f.: 6x/(1 - 10x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = 6*A004189(n). - R. J. Mathar, Jun 22 2020
6*a(n)^2+9 = (3*A001079(n))^2  - detail of the Barker comment. - R. J. Mathar, Jun 22 2020

More terms and better definition from Benoit Cloitre, Sep 23 2006

A072818 Possibly the only integers of the form sqrt(m^2(m^2-1)2/3) [only checked for the first 5 terms].

Original entry on oeis.org

0, 20, 1960, 192060, 18819920, 1844160100, 180708869880, 17707625088140, 1735166549767840, 170028614252160180, 16661069030161929800, 1632614736341616960220, 159979583092448300171760

Offset: 0

Henry Bottomley, Jul 14 2002

nonn

These are the standard deviations of time for a random walk starting at 0 to reach one of the boundaries at +A001079(n) or -A001079(n) for the first time.

			0 and 20 are at the start of the sequence since m^2*(m^2-1)*2/3 (A072819) starts 0, 0, 8, 48, 160, 400, 840, 1568, ... and the only squares among these are 0 and 400 with square roots of 0 and 20.

Indranil Ghosh, Table of n, a(n) for n = 0..501
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (98, -1).

a(n) = 98*a(n-1)-a(n-2) [starting with a(0)=0 and a(1)=20] =sqrt(A072819(A001079(n))).

G.f.: 20x/(1-98x+x^2). [Philippe Deléham, Nov 18 2008]

A098308 Unsigned member r=-8 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 8, 81, 800, 7921, 78408, 776161, 7683200, 76055841, 752875208, 7452696241, 73774087200, 730288175761, 7229107670408, 71560788528321, 708378777612800, 7012226987599681, 69413891098384008, 687126683996240401

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-8}(n), n>=0, defined in A092184.

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,9,-1).

LinearRecurrence[{9,9,-1},{0,1,8},40] (* Harvey P. Dale, Aug 11 2013 *)

a(n)= (T(n, 5)-(-1)^n)/6, with Chebyshev's polynomials of the first kind evaluated at x=5: T(n, 5)=A001079(n)=((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.
a(n)= 10*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 9*a(n-1) + 9*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=8.
G.f.: x*(1-x)/((1+x)*(1-10*x+x^2)) = x*(1-x)/(1-9*x-9*x^2+x^3) (from the Stephan link, see A092184).
a(x)=1/12(2*(-1)^x+(5+2*Sqrt[6])(5-2*Sqrt[6])^x+(5-2*Sqrt[6])(5+2*Sqrt[6])^x). - Harvey P. Dale, Aug 11 2013
a(n-1)+a(n) = A072256(n). - R. J. Mathar, Feb 19 2017

A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

Original entry on oeis.org

7, 799, 78407, 7683199, 752875207, 73774087199, 7229107670407, 708378777612799, 69413891098384007, 6801852948864019999, 666512175097575576007, 65311391306613542428799, 6399849835873029582446407, 627119972524250285537319199, 61451357457540654953074835207

Offset: 1

Bruno Berselli, Nov 23 2016

It is well known that T(m) + k*T(m+1) is always a square for k=1. For k=3, the nonnegative values of m are the terms of A278310.
Square roots of T(m) + 2*T(m+1) are listed by A168520 (after 0).
Negative values of m for which T(m) + 2*T(m+1) is a square: -1, -2, -82, -7922, -776162, ...

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Subsequence of A056220.
Cf. A000217, A001078, A001079, A122652, A168520, A278620.
Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.

Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];

P:=proc(q) local n; for n from 1 to q do if type(sqrt((3*n^2+7*n+4)/2),integer) then print(n); fi; od; end: P(10^9); #  Paolo P. Lava, Nov 25 2016

Table[((5 + 2 Sqrt[6])^(2 n) + (5 - 2 Sqrt[6])^(2 n))/12 - 7/6, {n, 1, 20}]
RecurrenceTable[{a[1] == 7, a[2] == 799, a[n] == 98 a[n - 1] - a[n - 2] + 112}, a, {n, 1, 20}]
LinearRecurrence[{99,-99,1},{7,799,78407},20] (* Harvey P. Dale, Oct 18 2024 *)

Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 27 2016

def A278438():
    a, b = 7, 799
    yield a
    while True:
        yield b
        a, b = b, 98*b - a + 112
a = A278438(); print([next(a) for  in range(15)]) # _Peter Luschny, Nov 24 2016

O.g.f.: x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: (exp((5-2*sqrt(6))^2*x) + exp((5+2*sqrt(6))^2*x) - 14*exp(x))/12 + 1.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>3.
a(n) = 98*a(n-1) - a(n-2) + 112 for n>2.
a(n) = a(-n) = ((5 + 2*sqrt(6))^(2*n) + (5 - 2*sqrt(6))^(2*n))/12 - 7/6.
a(n) = A001079(2*n)/6 - 7/6.
a(n) = 2*A001078(n)^2 - 1 = A122652(n)^2/2 - 1.
a(n) = -A278620(n+1) + 106*A278620(n) + 7*A278620(n-1).
Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.

A106330 Numbers k such that k^2 = 24*j^2 + 25.

Original entry on oeis.org

5, 7, 11, 25, 59, 103, 245, 583, 1019, 2425, 5771, 10087, 24005, 57127, 99851, 237625, 565499, 988423, 2352245, 5597863, 9784379, 23284825, 55413131, 96855367, 230496005, 548533447, 958769291, 2281675225, 5429921339, 9490837543, 22586256245, 53750679943

Offset: 1

Pierre CAMI, Apr 29 2005

The ratio k(n) /(2*j(n)) tends to sqrt(6) as n increases.

k(n) = 2*b + 1, for n > 0, where b is a side of the Heronian triangle (5, b, b+1). - Andrés Ventas, Dec 13 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-1).

Cf. A106331.

Vec(-x*(7*x^5+11*x^4+25*x^3-11*x^2-7*x-5)/(x^6-10*x^3+1) + O(x^100)) \\ Colin Barker, Apr 16 2014

Recurrence: k(1)=5, k(2)=7, k(3)=11, k(4)=25, k(5)=10*k(2)-k(3), k(6)=10*k(3)-k(2) then k(n)=10*k(n-3)-k(n-6).
From Ralf Stephan, Nov 15 2010: (Start)
G.f.: (-7x^5-11x^4-25x^3+11x^2+7x+5)/(x^6-10x^3+1).
a(3n+1) = 5*A001079(n), a(3n+2) = A077409(n), a(3n+3) = A077250(n). (End)

More terms from Ralf Stephan, Nov 15 2010

More terms from Colin Barker, Apr 16 2014

A290284 Number of pairs of integers (x,y) satisfying the Diophantine equation x^2 - A000037(n)*y^2 = m such that x/y gives a convergent series towards sqrt(A000037(n)).

Original entry on oeis.org

3, 3, 5, 4, 5, 4, 7, 6, 5, 15, 8, 5, 9, 7, 12, 6, 10, 12, 9, 6, 11, 9, 12, 21, 7, 17, 9, 10, 11, 7, 13, 10, 9, 9, 19, 8, 20, 15, 13, 24, 12, 8, 15, 12, 16, 27, 16, 13, 9, 14, 27, 17, 12

Offset: 1

A.H.M. Smeets, Jul 25 2017

If (x(0),y(0)) and (x(1),y(1)) are solutions of the Diophantine equation x^2 - A000037(n)*y^2 = m, then (x(i),y(i)) with x(i) = A*x(i-1) - x(i-2) and y(i) = A*y(i-1) - y(i-2) are also solutions for i > 1. The sequence represents the number of different integer pair sequences where in all cases A = 2*A033313(A000037(n)). Each contributing sequences has to satisfy the condition that for all x < x(i) and all y < y(i), |x/y - sqrt(A000037(n))| > |x(i)/y(i) - sqrt(A000037(n))|.
a(A000037(n)) is not equal to the number of all sequences of pairs (x(i),y(i)) that are solutions of a Diophantine equation  x^2 - D*y^2 = m, with -D <= m < D and D = A000037(n). For example for D = 5 we obtain two other sequences from Fibonacci sequence: (first) x(i) = 2*Fib(6i)-Fib(6i-1) and y(i) = Fib(6i-1) satisfy x^2 - D*y^2 = -4 and (second) x(i) = 2*Fib(6i+3) - Fib(6i+2) and y(i) = Fib(6i+2) satisfy x^2 - D*y^2 = 4; but neither of these satisfy the restriction that, for all x < x(i) and all y < y(i), |x/y - sqrt(D)| > |x(i)/y(i) - sqrt(D)|.
A good approximation for the order of magnitude of a(n) is given by 2*log(2*A033313(n)).
For a lower bound, all values m satisfying either m = -D + k^2 for k^2 < D or m = 1, D = A000037(n), contribute with a sequence to the convergent series of sqrt(D), so a(n) > floor(sqrt(D)) + 1.

			For A000037(4) = 6, a(4) = 4 we have the following sequences of pairs (x,y):
m = 1: x(0) = 1, x(1) = 5, x(i) = 10*x(i-1) - x(i-2) as in A001079(i) and y(0) = 0, y(1) = 2, y(i) = 10*y(i-1) - y(i-2) as in A001078(i);
m = -6: x(0) = 0, x(1) = 12, x(i) = 10*x(i-1) - x(i-2) as in A004291(i) (for i > 0) and y(0) = 1, y(1) = 5, y(i) = 10*y(i-1) - y(i-2) as in A001079(i);
m = -5: x(0) = 1, x(1) = 17, x(i) = 10*x(i-1) - x(i-2) and y(0) = 1, y(1) = 7, y(i) = 10*y(i-1) - y(i-2);
m = -2: x(0) = 2, x(1) = 22, x(i) = 10*x(i-1) - x(i-2) and y(0) = 1, y(1) = 9, y(i) = 10*y(i-1) - y(i-2) as in A072256(i+1).
In some cases a combination of A000037(n) and m has more than one integer pair sequence, for example A000037(5) = 7 and m = -3 has two integer pair sequences:
x(0) = 2, x(1) = 37, x(i) = 16*x(i-1) - x(i-2) and y(0) = 1, y(1) = 14, y(i) = 16*y(i-1) - y(i-2);
x(0) = -2, x(1) = 5, x(i) = 16*x(i-1) - x(i-2) and y(0) = 1, y(1) = 2, y(i) = 16*y(i-1) - y(i-2).
For A000037(4) = 6, the sequence observed from x^2 - 6y^2 = 3 is not in the convergent series of sqrt(6) due to for example x1/y1 = 2643/1079 = sqrt(6) + 5.259842e-7 while the smaller x,y pair, x2/y2 = 2158/881 from x^2 - 6y^2 = -2 is a fraction closer to sqrt(5), 2158/881 = sqrt(6) - 5.259841e-7.

Cf. A001078, A001079, A004291, A033313, A072256.

from fractions import Fraction
def FracSqrt(p):
    a = Fraction(p/1)
    b = Fraction(1/1)
    e = Fraction(10**(-200))
    while a-b > e:
        a = (a+b)/2
        b = p/a
    return a
print("number: ")
pp = int(input())
p = FracSqrt(pp)
n = 0
while n >= 0:
    n = n+1
    q = p.limit_denominator(n)
    if (n == 1) or (q != q0):
        t = q*n
        m = t*t-pp*n*n
        print(n,q,m)
    q0 = q

cp's OEIS Frontend

A165293 Inverse of A038303, and generalization of A130595.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

Formula

A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

PARI

Formula

Extensions

A072818 Possibly the only integers of the form sqrt(m^2*(m^2-1)*2/3) [only checked for the first 5 terms].

Views

Author

Keywords

Comments

Examples

Links

Formula

A098308 Unsigned member r=-8 of the family of Chebyshev sequences S_r(n) defined in A092184.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A106330 Numbers k such that k^2 = 24*j^2 + 25.

Views

Author

Keywords

A072818 Possibly the only integers of the form sqrt(m^2(m^2-1)2/3) [only checked for the first 5 terms].