cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A261004 Expansion of (-3-164*x-x^2)/(1-99*x+99*x^2-x^3).

Original entry on oeis.org

-3, -461, -45343, -4443321, -435400283, -42664784581, -4180713488823, -409667257120241, -40143210484294963, -3933624960203786301, -385455102889486762703, -37770666458209498958761, -3701139857801641411196043, -362673935398102648798253621, -35538344529156257940817658983
Offset: 0

Views

Author

N. J. A. Sloane, Aug 12 2015

Keywords

Comments

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 a_k.

Links

Crossrefs

Cf. A051028, A051029, A051030.
Cf. A010701, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

Programs

  • Mathematica
    LinearRecurrence[{99,-99,1},{-3,-461,-45343},30] (* Harvey P. Dale, Dec 02 2017 *)
  • PARI
    Vec((-3-164*x-x^2)/(1-99*x+99*x^2-x^3) + O(x^20)) \\ Michel Marcus, Feb 29 2016

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

Original entry on oeis.org

-1, -233, -22961, -2250073, -220484321, -21605213513, -2117090440081, -207453257914553, -20328302185186241, -1991966160890337193, -195192355465067858801, -19126858869415759825433, -1874236976847279395033761, -183656096872163964953483273, -17996423256495221286046327121
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

Cf. A010701, A261004, A269549, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    Vec((-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))
  • Sage
    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

Formula

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 + 298*x - 1)/(x^3 - 99*x^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

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

Cf. A010701, A261004, A269548, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    Vec((-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))
  • Sage
    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

Formula

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

Original entry on oeis.org

7, 465, 45347, 4443325, 435400287, 42664784585, 4180713488827, 409667257120245, 40143210484294967, 3933624960203786305, 385455102889486762707, 37770666458209498958765, 3701139857801641411196047, 362673935398102648798253625, 35538344529156257940817658987
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    LinearRecurrence[{99, -99, 1}, {7, 465, 45347}, 20] (* Vincenzo Librandi, Feb 29 2016 *)
  • PARI
    Vec((-5*x^2 + 228*x - 7)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

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

Original entry on oeis.org

3, 203, 19803, 1940403, 190139603, 18631740603, 1825720439403, 178901971320803, 17530567468999203, 1717816709990601003, 168328507011609899003, 16494475870427779501203, 1616290306794910781218803, 158379955590030828779941403, 15519619357516226309653038603, 1520764317081000147517217841603
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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

Original entry on oeis.org

-3, -435, -42763, -4190475, -410623923, -40236954115, -3942810879483, -386355229235355, -37858869654185443, -3709782870880938195, -363520862476677757803, -35621334739843539326635, -3490527283642190176252563, -342036052462194793733424675, -33516042614011447595699365723
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    LinearRecurrence[{99, -99, 1}, {-3, -435, -42763}, 20] (* Paolo Xausa, Mar 04 2024 *)
  • PARI
    Vec((-5*x^2 + 138*x + 3)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

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

Original entry on oeis.org

-1, -343, -33861, -3318283, -325158121, -31862177823, -3122168268781, -305940628162963, -29979059391701841, -2937641879758617703, -287858925156952833301, -28207237023501619046043, -2764021369378001713679161, -270845886962020666321511983, -26540132900908647297794495421
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    Vec((3*x^2 + 244*x + 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))
  • Sage
    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

Formula

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 + 254*x - 7)/(x^3 - 99*x^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

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    Vec((x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))
  • Sage
    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

Formula

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

Original entry on oeis.org

5, 347, 33865, 3318287, 325158125, 31862177827, 3122168268785, 305940628162967, 29979059391701845, 2937641879758617707, 287858925156952833305, 28207237023501619046047, 2764021369378001713679165, 270845886962020666321511987, 26540132900908647297794495425, 2600662178402085414517539039527
Offset: 0

Views

Author

Michel Marcus, Feb 29 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    Vec((-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))
  • Sage
    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

Formula

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

A386550 Indices of hexagonal numbers that are six times another hexagonal number.

Original entry on oeis.org

0, 2, 176, 17222, 1687556, 165363242, 16203910136, 1587817830062, 155589943435916, 15246226638889682, 1493974620667752896, 146394266598800894102, 14345144152061819869076, 1405677732635459546275322, 137742072654122973715112456, 13497317442371415964534745342
Offset: 1

Views

Author

Kelvin Voskuijl, Jul 25 2025

Keywords

Examples

			176 is in this sequence because the 176th hexagonal number (61776) is six times another hexagonal number.

Links

Crossrefs

Cf. A000384, A269551, A385917, A386549.

Formula

a(n) = (3*A269551(n-2) - 7)/4 for n>=2. - Hugo Pfoertner, Jul 26 2025
G.f.: 2*x^2*(1 - 11*x - 2*x^2)/((1 - x)*(1 - 98*x + x^2)). - Stefano Spezia, Jul 27 2025

Extensions

More terms from Jinyuan Wang, Jul 26 2025
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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