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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A005610 Number of Boolean functions realized by cascades of n gates.

Original entry on oeis.org

2, 14, 86, 518, 3110, 18662, 111974, 671846, 4031078, 24186470, 145118822, 870712934, 5224277606, 31345665638, 188073993830, 1128443962982, 6770663777894, 40623982667366, 243743896004198, 1462463376025190
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003464.

Programs

  • Maple
    A005610:=-2*(-7+6*z)/(6*z-1)/(z-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives the sequence apart from the initial 2
  • Mathematica
    Table[(2/5) (6 6^(n - 1) - 1), {n, 1, 30}] (* Bruno Berselli, Nov 13 2018 *)

Formula

a(n) = (2/5)*(6*6^(n-1) - 1). - Ralf Stephan, Apr 19 2004
a(n) = 6*a(n-1) + 2 for n > 1. - Georg Fischer, Nov 13 2018

Extensions

More terms from Ralf Stephan, Apr 19 2004

A016200 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-6*x)).

Original entry on oeis.org

1, 9, 61, 381, 2317, 13965, 83917, 503757, 3023053, 18139341, 108838093, 653032653, 3918204109, 23509241037, 141055478989, 846332939469, 5077997767885, 30467986869453, 182807921741005, 1096847531494605, 6581085191064781, 39486511150582989, 236919066911886541, 1421514401488096461
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A003464, A016129, A241275.

Programs

  • GAP
    List([0..100],n->(9*6^n-5*2^n+1)/5); # Muniru A Asiru, Feb 06 2018
  • Maple
    seq((9*6^n-5*2^n+1)/5, n=0..100); # Muniru A Asiru, Feb 06 2018
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{9,-20,12},{1,9,61},30] (* Harvey P. Dale, Aug 23 2025 *)

Formula

a(n) = (9*6^n - 5*2^n + 1)/5. - Bruno Berselli, Feb 09 2011
a(0)=1, a(n) = 6*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
a(n) = Sum_{k=0..n} 2^(n-1-k)*(3^(n+1-k) - 1). - J. M. Bergot, Feb 06 2018
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(9*exp(5*x) - 5*exp(x) + 1)/5.
a(n) = A016129(n+1) - A003464(n+2) = A241275(n+2)/5.
a(n) = 9*a(n-1) - 20*a(n-2) + 12*a(n-3). (End)

A024346 Expansion of 1/((1-x)*(1-6*x)*(1-9*x)*(1-11*x)).

Original entry on oeis.org

1, 27, 484, 7266, 98959, 1269177, 15642586, 187539120, 2204181925, 25529358855, 292445725936, 3321943348542, 37489352241979, 420930326166741, 4707254688375814, 52473555698990412, 583456285162491601
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A003464, A016244, A024347.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 25); Coefficients(R!(1/((1-x)*(1-6*x)*(1-9*x)*(1-11*x)))); // Vincenzo Librandi, Jul 16 2013
  • Magma
    I:=[1,27,484,7266]; [n le 4 select I[n] else 27*Self(n-1)-245*Self(n-2)+813*Self(n-3)-594*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Jul 16 2013
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-6x)(1-9x)(1-11x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 16 2013 *)
LinearRecurrence[{27,-245,813,-594},{1,27,484,7266},20] (* Harvey P. Dale, Oct 13 2016 *)
  • PARI
    a(n) = (12*11^(n+3) - 25*9^(n+3) + 16*6^(n+3) - 3)/1200; \\ Joerg Arndt, Aug 13 2013
  • Sage
    [(4*11^(n+3) -75*9^(n+2) +32*6^(n+2) -1)/400 for n in (0..20)] # G. C. Greubel, Jan 30 2022

Formula

a(n) = 27*a(n-1) - 245*a(n-2) + 813*a(n-3) - 594*a(n-4) for n > 3; a(0)=1, a(1)=27, a(2)=484, a(3)=7266. - Vincenzo Librandi, Jul 16 2013
a(n) = (12*11^(n+3) - 25*9^(n+3) + 16*6^(n+3) - 3)/1200. - Yahia Kahloune, Aug 13 2013
E.g.f.: (1/400)*(-exp(x) + 1152*exp(6*x) - 6075*exp(9*x) + 5324*exp(11*x)). - G. C. Greubel, Jan 30 2022

A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

Original entry on oeis.org

0, 6, 42, 258, 1554, 9330, 55986, 335922, 2015538, 12093234, 72559410, 435356466, 2612138802, 15672832818, 94036996914, 564221981490, 3385331888946, 20311991333682, 121871948002098, 731231688012594, 4387390128075570, 26324340768453426, 157946044610720562
Offset: 0

Views

Author

Alexandre Wajnberg, Apr 25 2005

Keywords

Comments

Number of integers from 0 to (10^n) - 1 that lack 0, 1, 2 and 3 as a digit.
a(n) is the expected number of tosses of a single die needed to obtain for the first time a string of n consecutive 6's. - Jean M. Morales, Aug 04 2012

Links

Crossrefs

Cf. A000918, A003464, A029858, A052379, A052386, A080674.
Row n=6 of A228275.

Programs

  • Maple
    a:=n->add(6^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Oct 03 2007
  • Mathematica
    NestList[6#+6&,0,30] (* Harvey P. Dale, Jul 24 2012 *)
  • PARI
    a(n)=if(n<0,0, (6^n-1)*6/5)

Formula

a(n) = 6^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jan 28 2013: (Start)
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 6*x/((x-1)*(6*x-1)). (End)
From Elmo R. Oliveira, Mar 16 2025: (Start)
E.g.f.: 6*exp(x)*(exp(5*x) - 1)/5.
a(n) = 6*(6^n - 1)/5.
a(n) = 6*A003464(n). (End)

Extensions

More terms from Harvey P. Dale, Jul 24 2012

A218727 a(n) = (24^n - 1)/23.

Original entry on oeis.org

0, 1, 25, 601, 14425, 346201, 8308825, 199411801, 4785883225, 114861197401, 2756668737625, 66160049703001, 1587841192872025, 38108188628928601, 914596527094286425, 21950316650262874201, 526807599606308980825, 12643382390551415539801, 303441177373233972955225
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 24 (A009968); q-integers for q=24: diagonal k=1 in triangle A022188.
Partial sums are in A014913. Also, the sequence is related to A014942 by A014942(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. [Bruno Berselli, Nov 07 2012]

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009968, A014942, A022188.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-24*x)).
a(n) = floor(24^n/23).
a(n) = 25*a(n-1) - 24*a(n-2). (End)
E.g.f.: exp(x)*(exp(23*x) - 1)/23. - Elmo R. Oliveira, Aug 29 2024

A218729 a(n) = (26^n - 1)/25.

Original entry on oeis.org

0, 1, 27, 703, 18279, 475255, 12356631, 321272407, 8353082583, 217180147159, 5646683826135, 146813779479511, 3817158266467287, 99246114928149463, 2580398988131886039, 67090373691429037015, 1744349715977154962391, 45353092615406029022167, 1179180408000556754576343
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 26 (A009970); q-integers for q=26.

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009970.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-26*x)).
a(n) = floor(26^n/25).
a(n) = 27*a(n-1) - 26*a(n-2). (End)
E.g.f.: exp(x)*(exp(25*x) - 1)/25. - Elmo R. Oliveira, Aug 29 2024

A218730 a(n) = (27^n - 1)/26.

Original entry on oeis.org

0, 1, 28, 757, 20440, 551881, 14900788, 402321277, 10862674480, 293292210961, 7918889695948, 213810021790597, 5772870588346120, 155867505885345241, 4208422658904321508, 113627411790416680717, 3067940118341250379360, 82834383195213760242721, 2236528346270771526553468
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 27 (A009971); q-integers for q=27.

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009971.

Programs

  • Magma
    [n le 2 select n-1 else 28*Self(n-1)-27*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
  • Mathematica
    LinearRecurrence[{28, -27}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
  • Maxima
    A218730(n):=(27^n-1)/26$
makelist(A218730(n),n,0,30); /* Martin Ettl, Nov 07 2012 */
  • PARI
    a(n)=27^n\26

Formula

G.f.: x/((1-x)*(1-27*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = floor(27^n/26). - Vincenzo Librandi, Nov 07 2012
a(n) = 28*a(n-1) - 27*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(14*x)*sinh(13*x)/13. - Elmo R. Oliveira, Aug 27 2024

A218731 a(n) = (28^n - 1)/27.

Original entry on oeis.org

0, 1, 29, 813, 22765, 637421, 17847789, 499738093, 13992666605, 391794664941, 10970250618349, 307167017313773, 8600676484785645, 240818941573998061, 6742930364071945709, 188802050194014479853, 5286457405432405435885, 148020807352107352204781, 4144582605859005861733869
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 28 (A009972).

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009972.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-28*x)).
a(n) = floor(28^n/27).
a(n) = 29*a(n-1) - 28*a(n-2). (End)
E.g.f.: exp(x)*(exp(27*x) - 1)/27. - Elmo R. Oliveira, Aug 29 2024

A218739 a(n) = (36^n - 1)/35.

Original entry on oeis.org

0, 1, 37, 1333, 47989, 1727605, 62193781, 2238976117, 80603140213, 2901713047669, 104461669716085, 3760620109779061, 135382323952046197, 4873763662273663093, 175455491841851871349, 6316397706306667368565, 227390317427040025268341, 8186051427373440909660277
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 36 (A009980).

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009980.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 36*x)).
a(n) = 37*a(n-1) - 36*a(n-2).
a(n) = floor(36^n/35). (End)
E.g.f.: exp(x)*(exp(35*x) - 1)/35. - Stefano Spezia, Mar 28 2023

A218741 a(n) = (38^n - 1)/37.

Original entry on oeis.org

0, 1, 39, 1483, 56355, 2141491, 81376659, 3092313043, 117507895635, 4465300034131, 169681401296979, 6447893249285203, 245019943472837715, 9310757851967833171, 353808798374777660499, 13444734338241551098963, 510899904853178941760595, 19414196384420799786902611
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 38 (A009982).

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A009982.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-38*x)).
a(n) = 39*a(n-1) - 38*a(n-2).
a(n) = floor(38^n/37). (End)
E.g.f.: exp(x)*(exp(37*x) - 1)/37. - Elmo R. Oliveira, Aug 29 2024
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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