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 11 results. Next

A099970 Write 1/e as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1. Then convert those integers from binary into decimal numbers.

Original entry on oeis.org

1, 5, 13, 29, 61, 573, 2621, 6717, 23101, 88637, 350781, 875069, 9263677, 26040893, 93149757, 227367485, 2374851133, 10964785725, 28144654909, 165583608381, 440461515325, 990217329213, 3189240584765, 7587287095869, 16383380118077
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

Keywords

Examples

			1/e = 0.367879441171442321595523770161460867445811131031767834507... = 0.010111100010110101011000110110001011001110111100110111110001101010111010110111 in binary.

Links

Crossrefs

Cf. A068985, A099969, A099971, A099972, A099973, A099974.

Programs

  • Mathematica
    d = 100; l = First[RealDigits[N[1/E, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)
Module[{nn=50,e},e=RealDigits[1/E,2, 50][[1]];Table[If[e[[n]]== 0, Nothing,FromDigits[ Reverse[Take[e,n]],2]],{n,nn}]] (* Harvey P. Dale, Sep 17 2020 *)

Formula

a(n) = A099969(n)/2. - Michel Marcus, Nov 03 2013

Extensions

More terms from Ryan Propper, Aug 18 2005
Definition amended by Harvey P. Dale, Sep 17 2020

A099969 Write 1/e as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

2, 10, 26, 58, 122, 1146, 5242, 13434, 46202, 177274, 701562, 1750138, 18527354, 52081786, 186299514, 454734970, 4749702266, 21929571450, 56289309818, 331167216762, 880923030650, 1980434658426, 6378481169530, 15174574191738
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

Keywords

Examples

			1/e = 0.367879441171442321595523770161460867445811131031767834507... = 0.010111100010110101011000110110001011001110111100110111110001101010111010110111 in binary.
From the binary expansion we get 10 = 2, 1010 = 10, 11010 = 26, 111010 = 58, 1111010 = 122, etc.

Crossrefs

Cf. A068985, A099970, A099971, A099972, A099973, A099974.

Programs

  • Mathematica
    d = 100; l = First[RealDigits[N[1/E, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[2*FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

Extensions

More terms from Ryan Propper, Aug 18 2005

A099971 Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 9, 25, 57, 121, 1145, 3193, 11385, 27769, 60537, 191609, 453753, 978041, 2026617, 10415225, 27192441, 94301305, 228519033, 496954489, 2644438137, 11234372729, 28414241913, 62773980281, 131493457017, 268932410489, 543810317433
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

Keywords

Examples

			(sqrt(5)-1)/2 = 0.618033988749894848204586834365638117720309179805762862135... = 0.100111100011011101111001101110010111111101001010011111000001010111110011... in binary.

Crossrefs

Cf. A094214, A099969, A099970, A099972, A099973, A099974.

Programs

  • Mathematica
    d = 100; l = First[RealDigits[N[(Sqrt[5]-1)/2, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

Extensions

More terms from Ryan Propper, Aug 18 2005

A099972 Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 5, 13, 45, 173, 8365, 73901, 204973, 467117, 991405, 5185709, 13574317, 80683181, 214900909, 1288642733, 3436126381, 7731093677, 16321028269, 33500897453, 67860635821, 136580112557, 686335926445, 1785847554221
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

Keywords

Examples

			1/sqrt(2) = 0.7071067811865475244008443621048490392848359376885... = 0.10110101000001001111001100110011111110011101111001100100100001000101100101111101100010011011 in binary.

Links

Crossrefs

Cf. A010503, A099969, A099970, A099971, A099973, A099974.

Programs

  • Mathematica
    d = 100; l = First[RealDigits[N[1/Sqrt[2], d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)
Module[{rd=RealDigits[1/Sqrt[2],2,50][[1]],pos},pos=Flatten[Position[rd,1]];Table[ FromDigits[ Reverse[Take[rd,n]],2],{n,pos}]] (* Harvey P. Dale, Jul 29 2013 *)

Extensions

More terms from Ryan Propper, Aug 18 2005

A099973 Write Euler's constant gamma as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 9, 73, 201, 457, 969, 9161, 140233, 402377, 2499529, 6693833, 15082441, 31859657, 65414089, 132522953, 1206264777, 3353748425, 11943683017, 29123552201, 63483290569, 132202767305, 269641720777, 819397534665
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

Keywords

Examples

			gamma = 0.577215664901532860606512090082402431042159335939923598805... = 0.1001001111000100011001111110001101111101101100001100011110100100110100011011... in binary.

Crossrefs

Cf. A001620, A099969, A099970, A099971, A099972, A099974.

Programs

  • Mathematica
    d = 100; l = First[RealDigits[N[EulerGamma, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

Extensions

More terms from Ryan Propper, Aug 18 2005

A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

Original entry on oeis.org

1, 5, 13, 45, 109, 365, 877, 2925, 7021, 23405, 56173, 187245, 449389, 1497965, 3595117, 11983725, 28760941, 95869805, 230087533, 766958445, 1840700269, 6135667565, 14725602157, 49085340525, 117804817261, 392682724205
Offset: 1

Views

Author

Artur Jasinski, Jan 27 2006

Keywords

Crossrefs

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.
Partial sums of A094015.

Formula

Empirical g.f.: x*(4*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Extensions

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113829 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.

Original entry on oeis.org

1, 9, 25, 57, 185, 441, 4537, 37305, 102841, 233913, 758201, 1806777, 18583993, 152801721, 421237177, 958108089, 3105591737, 7400559033, 76120035769, 625875849657, 1725387477433, 3924410732985, 12720503755193, 30312689799609
Offset: 1

Views

Author

Artur Jasinski, Jan 27 2006

Keywords

Links

Crossrefs

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971, A154571.

Programs

  • Mathematica
    LinearRecurrence[{1,0,0,0,0,4096,-4096},{1,9,25,57,185,441,4537},30] (* Harvey P. Dale, Aug 04 2018 *)
  • PARI
    Vec((-4096*x^6+4096*x^5+256*x^4+128*x^3+32*x^2+16*x+9)/(4096*x^7 - 4096*x^6-x+1)+O(x^99)) \\ Charles R Greathouse IV, Apr 05 2012

Formula

G.f.: (9+16*x+32*x^2+128*x^3+256*x^4+4096*x^5-4096*x^6)/(1-x-4096*x^6+4096*x^7). - Charles R Greathouse IV, Apr 05 2012

Extensions

Better definition, corrected offset and edited by Omar E. Pol, Jan 08 2009

A113841 a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.

Original entry on oeis.org

1, 3, 7, 71, 199, 455, 4551, 12743, 29127, 291271, 815559, 1864135, 18641351, 52195783, 119304647, 1193046471, 3340530119, 7635497415, 76354974151, 213793927623, 488671834567, 4886718345671, 13682811367879, 31274997412295
Offset: 1

Views

Author

Artur Jasinski, Jan 27 2006

Keywords

Links

Crossrefs

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 2 x + 4 x^2) / ((-1 + x) (-1 + 4 x) (1 + 4 x + 16 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 19 2013 *)
LinearRecurrence[{1,0,64,-64},{1,3,7,71},30] (* Harvey P. Dale, Nov 18 2013 *)

Formula

G.f.: x*(1+2*x+4*x^2)/((-1+x)*(-1+4*x)*(1+4*x+16*x^2)). - Vaclav Kotesovec, Nov 28 2012
a(1)=1, a(2)=3, a(3)=7, a(4)=71, a(n)=a(n-1)+64*a(n-3)-64*a(n-4). - Harvey P. Dale, Nov 18 2013

Extensions

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113867 a(n) = a(n-1) + 2^(A047258(n)) for n>1, a(1)=1.

Original entry on oeis.org

1, 17, 49, 113, 1137, 3185, 7281, 72817, 203889, 466033, 4660337, 13048945, 29826161, 298261617, 835132529, 1908874353, 19088743537, 53448481905, 122167958641, 1221679586417, 3420702841969, 7818749353073, 78187493530737
Offset: 1

Views

Author

Artur Jasinski, Jan 27 2006

Keywords

Links

Crossrefs

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 16 x + 32 x^2) / ((-1 + x) (- 1 + 4 x) (1 + 4 x + 16 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 20 2013 *)

Formula

G.f.: x*(1+16*x+32*x^2)/((-1+x)*(-1+4*x)*(1+4*x+16*x^2)). - Vaclav Kotesovec, Nov 28 2012

Extensions

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113870 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042963.

Original entry on oeis.org

1, 3, 7, 39, 103, 615, 1639, 9831, 26215, 157287, 419431, 2516583, 6710887, 40265319, 107374183, 644245095, 1717986919, 10307921511, 27487790695, 164926744167, 439804651111, 2638827906663, 7036874417767, 42221246506599
Offset: 1

Views

Author

Artur Jasinski, Jan 27 2006

Keywords

Links

Crossrefs

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Programs

Formula

G.f.: (3+x-40*x^2)/(4*(-1+x)*(-1+4*x)*(1+4*x)). - Vaclav Kotesovec, Nov 28 2012

Extensions

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009
Showing 1-10 of 11 results. Next

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

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