A120689 -id:A120689 - cp's OEIS frontend

A016131 Expansion of 1/((1-2x)(1-8*x)).

1, 10, 84, 680, 5456, 43680, 349504, 2796160, 22369536, 178956800, 1431655424, 11453245440, 91625967616, 733007749120, 5864062009344, 46912496107520, 375299968925696, 3002399751536640, 24019198012555264, 192153584100966400

Offset: 0

N. J. A. Sloane

"Numbral" powers of 10 (see A048888 for definition). - John W. Layman, Dec 18 2001
For n > 1, a(n-1) is the (2^n-3)rd coefficient in the expansion of th(0)=y, th(n+1) = th(n)*(th(n) + 1).
If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. Now consider the term 8^n-2^n: it must at least be divisible by 6 because it is the area of a PPT. a(n) is 1/6 the area of such triangles. - Frank M Jackson, Dec 28 2017

Iain Fox, Table of n, a(n) for n = 0..1107
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A000079, A002450, A016203, A048888, A059155, A120689, A248217.

[Binomial(2^(n+1)+1,3): n in [0..40]]; // G. C. Greubel, Dec 26 2024

seq(binomial(2^n,2)*(2^n + 1)/3,n=1..20); # Zerinvary Lajos, Jan 07 2008

CoefficientList[Series[1/((1 - 2x)(1 - 8x)), {x, 0, 100}], x] (* Stefan Steinerberger, Apr 21 2006 *)
a[n_] := (8^n-2^n)/6; Array[a, 20] (* Frank M Jackson, Dec 28 2017 *)

Vec(1/((1-2*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

def A016131(n): return binomial(pow(2,n+1) +1, 3)
print([A016131(n) for n in range(41)]) # G. C. Greubel, Dec 26 2024

[lucas_number1(n,10,16) for n in range(1, 21)] # Zerinvary Lajos, Apr 26 2009

[(8^n - 2^n)/6 for n in range(1,21)] # Zerinvary Lajos, Jun 05 2009

a(0) = 1, a(n) = (2^(3n+2) - 2^n)/3 = A059155(n)/12 = A000079(n)*A002450(n+1) = A016203(n+1) - A016203(n). - Ralf Stephan, Aug 14 2003
a(n) = binomial(2^n,2)*(2^n + 1)/3, n >= 1. - Zerinvary Lajos, Jan 07 2008
a(n) = (8^(n+1) - 2^(n+1))/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = Sum_{i=1...(2^n -1)} i*(i+1)/2. - Ctibor O. Zizka, Mar 03 2009
a(0) = 1, a(n) = 8*a(n-1) + 2^n. - Vincenzo Librandi, Feb 09 2011
a(n) = 10*a(n-1) - 16*a(n-2), n > 1. - Vincenzo Librandi, Feb 09 2011
a(n) = A248217(n+1)/6. - Frank M Jackson, Dec 28 2017
E.g.f.: e^(2*x) * (4*e^(6*x) - 1)/3. - Iain Fox, Dec 28 2017

A130213 Order of modular group of degree 3^(n-1) + 1.

Original entry on oeis.org

0, 12, 360, 9828, 265680, 7174332, 193709880, 5230175508, 141214764960, 3812798732652, 102945566017800, 2779530283189188, 75047317648233840, 2026277576508690972, 54709494565753788120, 1477156353275409674868, 39883221538436233408320, 1076846981537778818585292

Offset: 1

Artur Jasinski, Aug 04 2007

E. Mathieu, Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles, Journ. de math. (2) 5 (1860), 9-42 (see p. 39).

Ronald P. Nordgren, Compound Lucas Magic Squares, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
Index entries for linear recurrences with constant coefficients, signature (30,-81).

Cf. A120689.

Table[3^(x - 1) (3^(2 x - 2) - 1)/2, {x, 1, 15}]

a(n) = 3^(n-1)*(3^(2*n - 2) - 1)/2.
From Colin Barker, Sep 02 2013: (Start)
a(n) = 30*a(n-1) - 81*a(n-2).
G.f.: 12*x^2 / ((3*x-1)*(27*x-1)). (End)

More terms from Colin Barker, Sep 02 2013

A130214 Order of modular group of degree 5^(n-1)+1.

Original entry on oeis.org

0, 60, 7800, 976500, 122070000, 15258787500, 1907348625000, 238418579062500, 29802322387500000, 3725290298460937500, 465661287307734375000, 58207660913467382812500, 7275957614183425781250000, 909494701772928237304687500, 113686837721616029736328125000

Offset: 1

Artur Jasinski, Aug 04 2007

E. Mathieu, Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles, Journ. de math. (2) 5 (1860), 9-42 (see p. 39).

Index entries for linear recurrences with constant coefficients, signature (130,-625).

Cf. A120689.

Table[5^(x - 1) (5^(2 x - 2) - 1)/2, {x, 1, 15}]
LinearRecurrence[{130,-625},{0,60},30] (* Harvey P. Dale, Aug 09 2023 *)

a(n) = 5^(n - 1) (5^(2 n - 2) - 1)/2.

a(n) = 130*a(n-1)-625*a(n-2). G.f.: 60*x^2 / ((5*x-1)*(125*x-1)). - Colin Barker, Sep 02 2013

More terms from Colin Barker, Sep 02 2013

A130215 Order of modular group of degree 7^(n-1)+1.

Original entry on oeis.org

0, 168, 58800, 20176632, 6920642400, 2373780746568, 814206798896400, 279272932041230232, 95790615690280324800, 32856181181767119892968, 11269670145346128902694000, 3865496859853722261079883832, 1325865422929826735882591047200, 454771840064930570410054065439368

Offset: 1

Artur Jasinski, Aug 04 2007

E. Mathieu, Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles, Journ. de math. (2) 5 (1860), 9-42 (see p. 39).

Index entries for linear recurrences with constant coefficients, signature (350,-2401).

Cf. A120689.

Table[7^(x - 1) (7^(2 x - 2) - 1)/2, {x, 1, 15}]
LinearRecurrence[{350,-2401},{0,168},20] (* Harvey P. Dale, Aug 01 2022 *)

a(n) = (7^(n-1))*(7^(2n-2)-1)/2.

a(n) = 350*a(n-1)-2401*a(n-2). G.f.: 168*x^2 / ((7*x-1)*(343*x-1)). - Colin Barker, Sep 02 2013

More terms from Colin Barker, Sep 02 2013

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A016131 Expansion of 1/((1-2*x)*(1-8*x)).

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A130213 Order of modular group of degree 3^(n-1) + 1.

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A130214 Order of modular group of degree 5^(n-1)+1.

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A130215 Order of modular group of degree 7^(n-1)+1.

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A016131 Expansion of 1/((1-2x)(1-8*x)).