A261004 -id:A261004 - cp's OEIS frontend

A269551 Expansion of (3x^2 + 258x - 5)/(x^3 - 99x^2 + 99x - 1).

5, 237, 22965, 2250077, 220484325, 21605213517, 2117090440085, 207453257914557, 20328302185186245, 1991966160890337197, 195192355465067858805, 19126858869415759825437, 1874236976847279395033765, 183656096872163964953483277, 17996423256495221286046327125, 1763465823039659522067586574717

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 e_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, A269552, A269553, A269554, A269555, A269556.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3*x^2+258*x-5)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

CoefficientList[Series[(3 x^2 + 258 x - 5)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[8/3 + (-(3 Sqrt[6] - 7)/(2 Sqrt[6] + 5)^(2 n) + (3 Sqrt[6] + 7) (2 Sqrt[6] + 5)^(2 n))/6], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)

Vec((3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (3*x^2+258*x-5)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

G.f.: (3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 8/3 + (-(3*sqrt(6) - 7)/(2*sqrt(6) + 5)^(2*n) + (3*sqrt(6) + 7)*(2*sqrt(6) + 5)^(2*n))/6. - Bruno Berselli, Mar 01 2016

A269548 Expansion of (-7x^2 + 134x + 1)/(x^3 - 99x^2 + 99x - 1).

Original entry on oeis.org

-1, -233, -22961, -2250073, -220484321, -21605213513, -2117090440081, -207453257914553, -20328302185186241, -1991966160890337193, -195192355465067858801, -19126858869415759825433, -1874236976847279395033761, -183656096872163964953483273, -17996423256495221286046327121

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 b_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, A269549, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-7*x^2+134*x+1)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

CoefficientList[Series[(-7 x^2 + 134 x + 1)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[4/3 + ((3 Sqrt[6] - 7)/(2 Sqrt[6] + 5)^(2 n) - (3 Sqrt[6] + 7) (2 Sqrt[6] + 5)^(2 n))/6], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)

Vec((-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (-7*x^2+134*x+1)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

G.f.: (-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 4/3 + ((3*sqrt(6) - 7)/(2*sqrt(6) + 5)^(2*n) - (3*sqrt(6) + 7)*(2*sqrt(6) + 5)^(2*n))/6. - Bruno Berselli, Mar 01 2016

A269549 Expansion of (-x^2 + 298x - 1)/(x^3 - 99x^2 + 99*x - 1).

Original entry on oeis.org

1, -199, -19799, -1940399, -190139599, -18631740599, -1825720439399, -178901971320799, -17530567468999199, -1717816709990600999, -168328507011609898999, -16494475870427779501199, -1616290306794910781218799, -158379955590030828779941399, -15519619357516226309653038599

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 c_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, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-x^2+298*x-1)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

CoefficientList[Series[(-x^2 + 298 x - 1)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[37/12 + ((2 Sqrt[6] - 5)/(2 Sqrt[6] + 5)^(2 n) - (2 Sqrt[6] + 5) (2 Sqrt[6] + 5)^(2 n)) 5/24], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)

Vec((-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

gf = (-x^2+298*x-1)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

G.f.: (-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 37/12 + ((2*sqrt(6) - 5)/(2*sqrt(6) + 5)^(2*n) - (2*sqrt(6) + 5)*(2*sqrt(6) + 5)^(2*n))*5/24. - Bruno Berselli, Mar 01 2016

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

Original entry on oeis.org

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

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A269551 Expansion of (3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

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A269548 Expansion of (-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

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A269549 Expansion of (-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1).

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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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A269551 Expansion of (3x^2 + 258x - 5)/(x^3 - 99x^2 + 99x - 1).

A269548 Expansion of (-7x^2 + 134x + 1)/(x^3 - 99x^2 + 99x - 1).

A269549 Expansion of (-x^2 + 298x - 1)/(x^3 - 99x^2 + 99*x - 1).

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).