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.

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A013768 a(n) = 21^(3*n + 1).

Original entry on oeis.org

21, 194481, 1801088541, 16679880978201, 154472377739119461, 1430568690241985328321, 13248496640331026125580781, 122694327386105632949003612841, 1136272165922724266740722458520501, 10523016528610349434285830688358359761
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Subsequence of A009965.

Programs

A013898 a(n) = 21^(5*n + 1).

Original entry on oeis.org

21, 85766121, 350277500542221, 1430568690241985328321, 5842587018385982521381124421, 23861715484377209601555171628930521, 97453656071460446110921078004886769746621, 398010574215107679422058885600836061208944572721, 1625515384162495488635310116741260158419511738394408821
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) mod 10 == 1 (the least significant decimal digit of each term is 1).

Links

Crossrefs

Cf. A009965.

Programs

Formula

a(n) = 4084101*a(n-1), a(0)=21. - Vincenzo Librandi, May 27 2011

A013899 a(n) = 21^(5*n + 2).

Original entry on oeis.org

441, 1801088541, 7355827511386641, 30041942495081691894741, 122694327386105632949003612841, 501096025171921401632658604207540941, 2046526777500669368329342638102622164679041, 8358222058517261267863236597617557285387836027141
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A009965.

Programs

Formula

a(n) = 4084101*a(n-1), a(0)=441. - Vincenzo Librandi, May 27 2011

A013900 a(n) = 21^(5*n + 3).

Original entry on oeis.org

9261, 37822859361, 154472377739119461, 630880792396715529789561, 2576580875108218291929075869661, 10523016528610349434285830688358359761, 42977062327514056734916195400155065458259861, 175522663228862486625127968549968702993144556569961
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A009965.

Programs

Formula

a(n) = 4084101*a(n-1), a(0)=9261. - Vincenzo Librandi, May 27 2011

A013901 a(n) = 21^(5*n + 4).

Original entry on oeis.org

194481, 794280046581, 3243919932521508681, 13248496640331026125580781, 54108198377272584130510593262881, 220983347100817338120002444455525554981, 902518308877795191433240103403256374623457081, 3685975927806112219127687339549342762856035687969181
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A009965.

Programs

Formula

a(n) = 4084101*a(n-1), a(0)=194481. - Vincenzo Librandi, May 27 2011

A017968 Powers of sqrt(21) rounded to nearest integer.

Original entry on oeis.org

1, 5, 21, 96, 441, 2021, 9261, 42439, 194481, 891224, 4084101, 18715702, 85766121, 393029742, 1801088541, 8253624572, 37822859361, 173326116021, 794280046581, 3639848436450, 16679880978201, 76436817165460, 350277500542221, 1605173160474663, 7355827511386641
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010477 (sqrt(21)), A017967.
Bisection gives A009965 (even part).

Programs

  • Magma
    [Round(Sqrt(21)^n): n in [0..30]]; // Vincenzo Librandi, Nov 20 2011
  • Maple
    a:= n-> round(sqrt(21)^n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    Floor[(Sqrt[21])^Range[0,25]+1/2] (* Harvey P. Dale, Sep 22 2011 *)
  • PARI
    a(n)=round(sqrt(21)^n) \\ Charles R Greathouse IV, Nov 18 2011
  • Python
    from math import isqrt
def A017968(n): return (m:=isqrt(k:=21**n))+int((k-m*(m+1)<<2)>=1) # Chai Wah Wu, Jul 29 2022

A038335 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.

Original entry on oeis.org

1, 12, 9, 144, 216, 81, 1728, 3888, 2916, 729, 20736, 62208, 69984, 34992, 6561, 248832, 933120, 1399680, 1049760, 393660, 59049, 2985984, 13436928, 25194240, 25194240, 14171760, 4251528, 531441, 35831808, 188116992, 423263232
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			       1
      12        9
     144      216       81
    1728     3888     2916      729
   20736    62208    69984    34992     6561
  248832   933120  1399680  1049760   393660    59049
 2985984 13436928 25194240 25194240 14171760  4251528   531441

References

  • B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Crossrefs

Cf. A009965 (row sums), A001021 (column 0), A001019 (diagonal)

Programs

  • Maple
    A038335 := proc(i,j)
    binomial(i,j)*12^(i-j)*9^j ;
end proc: # R. J. Mathar, Nov 22 2022
  • Mathematica
    Flatten[Table[Binomial[i,j]12^(i-j) 9^j,{i,0,10},{j,0,i}]] (* Harvey P. Dale, Oct 17 2013 *)

A159705 Numerator of Hermite(n, 1/21).

Original entry on oeis.org

1, 2, -878, -5284, 2312620, 23267192, -10152119816, -143434219696, 62392319304592, 1136856492784160, -492996517654282976, -11013067301664857152, 4761026079678523718848, 126084356480177895534464, -54337756316633597169242240, -1665565146450503848398045952
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerator of 1, 2/21, -878/441, -5284/9261, 2312620/194481, 23267192/4084101, ...

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(2/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Maple
    A159705 := proc(n)
        orthopoly[H](n,1/21) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 17 2014
  • Mathematica
    Numerator[Table[HermiteH[n, 1/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n,1/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) - 2*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 17 2014
From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,1/21).
E.g.f.: exp(2*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/21)^(n-2k)/(k!*(n-2k)!). (End)

A159706 Numerator of Hermite(n, 2/21).

Original entry on oeis.org

1, 4, -866, -10520, 2249356, 46111984, -9735212024, -282965467424, 58973337166480, 2232497686809664, -459200359680279584, -21527431036382354816, 4369052165472543104704, 245322538750961015791360, -49114261974304335175370624, -3225699756394083963693195776
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerator of 1, 4/21, -866/441, -10520/9261, 2249356/194481, 46111984/4084101, ...

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(4/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 22 2018
  • Maple
    A159706 := proc(n)
        orthopoly[H](n,2/21) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 17 2014
  • Mathematica
    Numerator[Table[HermiteH[n, 2/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n,2/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) - 4*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 17 2014
From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,2/21).
E.g.f.: exp(4*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/21)^(n-2k)/(k!*(n-2k)!). (End)

A159707 Numerator of Hermite(n, 4/21).

Original entry on oeis.org

1, 8, -818, -20656, 1999180, 88867808, -8105441336, -535131970624, 45761939043472, 4141986697070720, -330122378550514976, -39173301696567870208, 2889460903124553335488, 437725912381470764965376, -29628751416174362424982400, -5642069577415795905192322048
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerator of 1, 8/21, -818/441, -20656/9261, 1999180/194481, 88867808/4084101, ...

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(8/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 22 2018
  • Maple
    A159707 := proc(n)
        orthopoly[H](n,4/21) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 17 2014
  • Mathematica
    Numerator[Table[HermiteH[n, 4/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n,4/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) - 8*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 17 2014
From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,4/21).
E.g.f.: exp(8*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(8/21)^(n-2k)/(k!*(n-2k)!). (End)
Previous Showing 11-20 of 32 results. Next

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

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