A010701 -id:A010701 - cp's OEIS frontend

A269550 Expansion of (-5x^2 + 228x - 7)/(x^3 - 99x^2 + 99x - 1).

7, 465, 45347, 4443325, 435400287, 42664784585, 4180713488827, 409667257120245, 40143210484294967, 3933624960203786305, 385455102889486762707, 37770666458209498958765, 3701139857801641411196047, 362673935398102648798253625, 35538344529156257940817658987

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence d_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A010701, A261004, A269548, A269549, A269551, A269552, A269553, A269554, A269555, A269556.

I:=[7,465,45347]; [n le 3 select I[n]  else 99*Self(n-1)+-99*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Feb 29 2016

LinearRecurrence[{99, -99, 1}, {7, 465, 45347}, 20] (* Vincenzo Librandi, Feb 29 2016 *)

Vec((-5*x^2 + 228*x - 7)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

A269552 Expansion of (-3x^2 + 94x - 3)/(x^3 - 99x^2 + 99x - 1).

Original entry on oeis.org

3, 203, 19803, 1940403, 190139603, 18631740603, 1825720439403, 178901971320803, 17530567468999203, 1717816709990601003, 168328507011609899003, 16494475870427779501203, 1616290306794910781218803, 158379955590030828779941403, 15519619357516226309653038603, 1520764317081000147517217841603

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence f_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99, -99, 1).

Cf. A010701, A261004, A269548, A269549, A269550, A269551, A269553, A269554, A269555, A269556.

CoefficientList[Series[(-3x^2+94x-3)/(x^3-99x^2+99x-1),{x,0,20}],x] (* or *) LinearRecurrence[{99,-99,1},{3,203,19803},20] (* Harvey P. Dale, Jan 14 2019 *)

Vec((-3*x^2 + 94*x - 3)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

G.f.: (-3*x^2 + 94*x - 3)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 99*a(n-1)-99*a(n-2)+a(n-3). - Wesley Ivan Hurt, May 20 2021

A269553 Expansion of (-5x^2 + 138x + 3)/(x^3 - 99x^2 + 99x - 1).

Original entry on oeis.org

-3, -435, -42763, -4190475, -410623923, -40236954115, -3942810879483, -386355229235355, -37858869654185443, -3709782870880938195, -363520862476677757803, -35621334739843539326635, -3490527283642190176252563, -342036052462194793733424675, -33516042614011447595699365723

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence p_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A010701, A261004, A269548, A269549, A269550, A269551, A269552, A269554, A269555, A269556.

LinearRecurrence[{99, -99, 1}, {-3, -435, -42763}, 20] (* Paolo Xausa, Mar 04 2024 *)

Vec((-5*x^2 + 138*x + 3)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

A269554 Expansion of (3x^2 + 244x + 1)/(x^3 - 99x^2 + 99x - 1).

Original entry on oeis.org

-1, -343, -33861, -3318283, -325158121, -31862177823, -3122168268781, -305940628162963, -29979059391701841, -2937641879758617703, -287858925156952833301, -28207237023501619046043, -2764021369378001713679161, -270845886962020666321511983, -26540132900908647297794495421

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence q_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A010701, A261004, A269548, A269549, A269550, A269551, A269552, A269553, A269555, A269556.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3*x^2+244*x+1)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 02 2016

CoefficientList[Series[(3 x^2 + 244 x + 1)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[Simplify[31/12 + ((17 Sqrt[6] - 43)/(2 Sqrt[6] + 5)^(2 n) - (17 Sqrt[6] + 43) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* Bruno Berselli, Mar 02 2016 *)
LinearRecurrence[{99,-99,1},{-1,-343,-33861},20] (* Harvey P. Dale, Feb 03 2025 *)

Vec((3*x^2 + 244*x + 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (3*x^2+244*x+1)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 02 2016

G.f.: (3*x^2 + 244*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 31/12 + ((17*sqrt(6) - 43)/(2*sqrt(6) + 5)^(2*n) - (17*sqrt(6) + 43)*(2 sqrt(6) + 5)^(2*n))/24. - Bruno Berselli, Mar 02 2016

A269555 Expansion of (x^2 + 254x - 7)/(x^3 - 99x^2 + 99*x - 1).

Original entry on oeis.org

7, 439, 42767, 4190479, 410623927, 40236954119, 3942810879487, 386355229235359, 37858869654185447, 3709782870880938199, 363520862476677757807, 35621334739843539326639, 3490527283642190176252567, 342036052462194793733424679, 33516042614011447595699365727, 3284230140120659669584804416319

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence r_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A010701, A261004, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269556.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^2+254*x-7)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

CoefficientList[Series[(x^2 + 254 x - 7)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[31/12 + (-(22 Sqrt[6] - 53)/(2 Sqrt[6] + 5)^(2 n) + (22 Sqrt[6] + 53) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)
LinearRecurrence[{99,-99,1},{7,439,42767},20] (* Harvey P. Dale, Apr 10 2019 *)

Vec((x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (x^2+254*x-7)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

G.f.: (x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 31/12 + (-(22*sqrt(6) - 53)/(2*sqrt(6) + 5)^(2*n) + (22*sqrt(6) + 53)*(2*sqrt(6)+5)^(2*n))/24. - Bruno Berselli, Mar 01 2016

A269556 Expansion of (-7x^2 + 148x - 5)/(x^3 - 99x^2 + 99x - 1).

Original entry on oeis.org

5, 347, 33865, 3318287, 325158125, 31862177827, 3122168268785, 305940628162967, 29979059391701845, 2937641879758617707, 287858925156952833305, 28207237023501619046047, 2764021369378001713679165, 270845886962020666321511987, 26540132900908647297794495425, 2600662178402085414517539039527

Offset: 0

Michel Marcus, Feb 29 2016

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence s_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A010701, A261004, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269555.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 02 2016

CoefficientList[Series[(-7 x^2 + 148 x - 5)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[Simplify[17/12 + (-(17 Sqrt[6] - 43)/(2 Sqrt[6] + 5)^(2 n) + (17 Sqrt[6] + 43) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* Bruno Berselli, Mar 02 2016 *)

Vec((-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 02 2016

G.f.: (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 17/12 + (-(17*sqrt(6) - 43)/(2*sqrt(6) + 5)^(2*n) + (17*sqrt(6) + 43)*(2 sqrt(6) + 5)^(2*n))/24. - Bruno Berselli, Mar 02 2016

A010703 Period 2: repeat (3,5).

Original entry on oeis.org

3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010701 and A010716.
Also continued fraction expansion of (15+sqrt(285))/10.
Also decimal expansion of 35/99.
Binomial transform of 3 followed by A084633 without initial terms 1,0.
Inverse binomial transform of A171497. (End)

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

Cf. A010701 (all 3's sequence), A010716 (all 5's sequence), A084633 (inverse binomial transform of repeated odd numbers), A171497.

&cat[ [3, 5]: n in [1..53] ]; // Klaus Brockhaus, Dec 10 2009

Table[If[OddQ[n], 3, 5], {n, 1, 50}] (* Stefan Steinerberger, Apr 10 2006 *)
PadRight[{},120,{3,5}] (* Harvey P. Dale, Sep 03 2012 *)

a(n)=3+n%2*2 \\ Charles R Greathouse IV, Nov 20 2011

From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 5.
G.f.: (3+5*x)/((1-x)*(1+x)). (End)
a(n) = 4 - (-1)^n. - Aaron J Grech, Aug 02 2024
E.g.f.: 3*cosh(x) + 5*sinh(x). - Stefano Spezia, Aug 04 2024

A021016 Decimal expansion of 1/12.

Original entry on oeis.org

0, 8, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

N. J. A. Sloane

Multiplied by -1, this is zeta(-1) or zeta(-13), with zeta being the Riemann zeta function. Divided by 10, this is zeta(-3). - Alonso del Arte, Jan 05 2011
Multiplied by 10, this is 5/6, the resistance in ohm between opposite vertices of a cubical network when each edge has a resistance of 1 ohm. - Michel Marcus, Sep 02 2015
The variance of a continuous uniform distribution U(a,b) is (1/12)*(b-a)^2. - Jean-François Alcover, May 19 2016
5/6 is the Schnirelmann density of the sums of three squares and also the asymptotic density of the set of sums of three squares. See Wagstaff. - Michel Marcus, Apr 22 2020
-1/12 = zeta(-1) is the Ramanujan sum of 1 + 2 + 3 + .... [see facsimile] and was called "one of the most remarkable formulae in science" [Gannon]. - Peter Luschny, Jul 17 2020

			0.083333333333333333333333333333333333333333333333333333333333333333...

Bruce C. Berndt, Ramanujan's Notebooks: Part 1, Springer-Verlag, 1985, pp. 135-136
Terry Gannon, Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, 2010, p. 140.
L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 40 (series n. 209) and p. 44 (series n. 239).

Martin Gardner, The Five Platonic Solids, Mathematical Puzzles & Diversions.
Srinivasa Ramanujan, Question 463, Journal of the Indian Mathematical Society, Vol. 5 (1913), p. 120.
Srinivasa Ramanujan, Another way of finding the constant, Notebook 1, 1919.
Samuel S. Wagstaff, Jr., The Schnirelmann Density of the Sums of Three Squares, Proc. Amer. Math. Soc. 52 (1975), 1-7.
Wikipedia, 1 + 2 + 3 + 4 + ....
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A005408 (odd numbers), A010701.

RealDigits[1/12, 10, 100, -1][[1]] (* Bruno Berselli, Mar 21 2014 *)

PARI
```
1/12. \\ Michel Marcus, Mar 11 2018
```

Equals 1/(1*3*5) + 1/(3*5*7) + 1/(5*7*9) + 1/(7*9*11) + ... = Sum_{i >= 0} 1/((2*i+1)*(2*i+3)*(2*i+5)), see Jolley in References. - Bruno Berselli, Mar 21 2014
Equals 1/(2*3*4) + 1/(3*4*5) + 1/(4*5*6) + 1/(5*6*7) + ... = Sum_{i > 0} 1/((i+1)*(i+2)*(i+3)). See Jolley in References, p. 48 (sum obtained from the series 268, case t = 2). - Bruno Berselli, Mar 29 2014
Equals 2*Pi*Integral_{z=-oo..oo} (z/(e^(-Pi*z) + e^(Pi*z)))^2. - Peter Luschny, Jul 17 2020
Equals lim_{x->oo} (P(x) - (1 - t(x))/(1 + t(x)))^(1/x) = lim_{x->oo} (t(x) - (1 - P(x))/(1 + P(x)))^(1/x) by the inversion, where P(x) is the prime zeta function of x and t(x) = zeta(2x)/zeta(x)^2, with zeta(x) being the Riemann zeta function of x. - Thomas Ordowski, Oct 28 2024
Equals Integral_{x>=0} 1/(exp(2*Pi*sqrt(x))-1) dx (Ramanujan, 1913). - Amiram Eldar, Jan 01 2025
Equals Integral_{x=0..1} x^(1/5) - x^(1/3) dx. - Kritsada Moomuang, May 27 2025

A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 1, 6, 6, 5, 6, 2, 8, 8, 6, 3, 8, 4, 10, 10, 7, 8, 1, 10, 6, 12, 12, 8, 5, 10, 2, 12, 8, 14, 14, 9, 10, 3, 12, 4, 14, 10, 16, 16, 10, 7, 12, 1, 14, 6, 16, 12, 18, 18, 11, 12, 5, 14, 2, 16, 8, 18, 14, 20, 20, 12, 9, 14, 3, 16, 4, 18, 10, 20, 16, 22, 22, 13, 14, 7, 16, 1

Offset: 0

Antti Karttunen, Oct 19 2001

Simple periodic site swap permutations of natural numbers.

Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...

			Table begins:
1 2 3 4 5 6 7 ...
2 4 1 6 3 8 5 ...
4 6 2 8 1 10 3 ...
6 8 4 10 2 12 1 ...

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Juggling Information Service, Site Swap FAQs

Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.

PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
[seq(PerSS_table(j),j=0..119)];

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).

A078890 Decimal expansion of Sum {n>=0} 1/9^(2^n).

Original entry on oeis.org

1, 2, 3, 6, 0, 9, 2, 2, 9, 1, 4, 4, 3, 0, 6, 3, 2, 7, 7, 8, 2, 1, 5, 8, 4, 9, 2, 8, 3, 3, 5, 6, 9, 7, 0, 4, 6, 5, 6, 6, 0, 9, 2, 6, 6, 3, 8, 0, 4, 9, 7, 8, 5, 8, 7, 5, 7, 7, 2, 3, 5, 4, 0, 9, 7, 6, 6, 4, 9, 0, 8, 4, 9, 6, 6, 3, 5, 6, 6, 9, 6, 1, 8, 5, 4, 9, 1, 8, 1, 9, 3, 3, 4, 7, 3, 5, 4, 4, 2, 0, 0, 1, 1, 9, 1

Offset: 0

Robert G. Wilson v, Dec 11 2002

			0.123609229144306327782...

Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Index entries for transcendental numbers

Similar sums: A007404, A078885, A078585, A078886, A078887, A078888, A078889, A036987.

RealDigits[ N[ Sum[1/9^(2^n), {n, 0, Infinity}], 110]][[1]]

suminf(n=0, 1/9^(2^n)) \\ Michel Marcus, Nov 11 2020

Equals A078885 - 1/3 = A078885 - A010701. - R. J. Mathar, Apr 23 2009

Equals -Sum_{k>=1} mu(2*k)/(9^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

cp's OEIS Frontend

A269550 Expansion of (-5*x^2 + 228*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

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A269552 Expansion of (-3*x^2 + 94*x - 3)/(x^3 - 99*x^2 + 99*x - 1).

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A269553 Expansion of (-5*x^2 + 138*x + 3)/(x^3 - 99*x^2 + 99*x - 1).

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A269554 Expansion of (3*x^2 + 244*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

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A269555 Expansion of (x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

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A269556 Expansion of (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

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A010703 Period 2: repeat (3,5).

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A269550 Expansion of (-5x^2 + 228x - 7)/(x^3 - 99x^2 + 99x - 1).

A269552 Expansion of (-3x^2 + 94x - 3)/(x^3 - 99x^2 + 99x - 1).

A269553 Expansion of (-5x^2 + 138x + 3)/(x^3 - 99x^2 + 99x - 1).

A269554 Expansion of (3x^2 + 244x + 1)/(x^3 - 99x^2 + 99x - 1).

A269555 Expansion of (x^2 + 254x - 7)/(x^3 - 99x^2 + 99*x - 1).

A269556 Expansion of (-7x^2 + 148x - 5)/(x^3 - 99x^2 + 99x - 1).