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 1-8 of 8 results.

A002064 Cullen numbers: a(n) = n*2^n + 1.

Original entry on oeis.org

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015
Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015
Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021
Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

Examples

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018

References

  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • R. K. Guy, Unsolved Problems in Number Theory, B20.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
  • W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Cf. A000325, A186947.
Diagonal k = n + 1 of A046688.
A000005 counts divisors of n.
A000312 = n^n.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
A173339 lists positions of squares in A062319.
A188385 gives the highest prime exponent in n^n.
A249784 counts divisors of n^n^n.
Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

Programs

Formula

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003
a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007
Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007
Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008
Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009
G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007
a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008
a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011
a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013
E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014
a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019
a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021
Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Extensions

Edited by M. F. Hasler, Oct 31 2012

A065246 Formal neural networks with n components.

Original entry on oeis.org

1, 4, 196, 1124864, 12545225621776, 7565068551396549351877632, 11519413104737198429297238164593057431690816, 3940200619639447921227904010014361380507973927046544666794829340424572177149721061141426654884915640806627990306816
Offset: 0

Views

Author

Labos Elemer, Oct 26 2001

Keywords

Comments

Number of {0,1}^n to {0,1}^n vector-vector maps of which all components are formal neurons (=threshold gates).

Examples

			For n=2 the 14 threshold gates determine 14*14=196 neural nets each built purely from threshold gates. For n=3, 104=A000609(3) formal neurons gives 104^3=a(3) networks, all component functions of which are linearly separable {0,1}^3 -> {0,1} vector-scalar functions.

References

  • Labos E. (1996): Long Cycles and Special Categories of Formal Neuronal Networks. Acta Biologica Hungarica, 47: 261-272.
  • Labos E. and Sette M. (1995): Long Cycle Generation by McCulloch-Pitts Networks(MCP-Nets) with Dense and Sparse Weight Matrices. Proc. of BPTM, McCulloch Memorial Conference [eds:Moreno-Diaz R. and Mira-Mira J., pp. 350-359.], MIT Press, Cambridge,MA,USA.
  • McCulloch, W. S. and Pitts W. (1943): A Logical Calculus Immanent in Nervous Activity. Bull. Math. Biophys. 5:115-133.

Crossrefs

Cf. A000609, A065247, A065248, A064436.

Formula

a(n)=A000609(n)^n; for n>1 a(n) < A057156(n).

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

References

  • G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Links

Crossrefs

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Programs

  • Mathematica
    Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)
  • PARI
    A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

Formula

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020
As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A057157 Number of non-invertible functions from {0,1}^n to {0,1}^n.

Original entry on oeis.org

0, 2, 232, 16736896, 18446723150919663616, 1461501637330639787366751139186115801643772542976
Offset: 0

Views

Author

Henry Bottomley, Aug 15 2000

Keywords

Links

Crossrefs

Cf. A057156, A000722, A036679, A000079.

Programs

  • Magma
    [(2^n)^(2^n)-Factorial(2^n): n in [0..5]]; // Vincenzo Librandi, Aug 22 2011
  • Mathematica
    Table[(2^n)^(2^n) - (2^n)!, {n,0,5}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(6,n,n--; (2^n)^(2^n) - (2^n)!) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = (2^n)^(2^n) - (2^n)! = A057156(n) - A000722(n) = A036679(A000079(n)).

A065247 Imperfect formal neural networks with n components.

Original entry on oeis.org

0, 0, 60, 15652352, 18446731528483929840, 1461501637330902918203677267647731623106580665344, 3940200619639447921227904010014361380507973
Offset: 0

Views

Author

Labos Elemer, Oct 26 2001

Keywords

Comments

Number of {0,1}^n to {0,1}^n vector-vector maps of which at least one component is not a formal neuron, i.e., some are not threshold gates.

Examples

			For n = 2 the 14 threshold gates determine 14*14 = 196 neural nets each built purely from threshold gates; the remaining 2^(2*4)-14^2 = 256-196 = 60 = a(2) functions are synthesized from both neurons and non-neurons. For n = 3, 104 = A000609(3) formal neurons and 152 non-neurons gives (2^24)-A065246(3) = 15652352 = a(4) nets with at least one linearly non-separable component.

References

  • Labos E. (1996): Long Cycles and Special Categories of Formal Neuronal Networks. Acta Biologica Hungarica, 47: 261-272.
  • Labos E. and Sette M.(1995): Long Cycle Generation by McCulloch-Pitts Networks(MCP-Nets) with Dense and Sparse Weight Matrices. Proc. of BPTM, McCulloch Memorial Conference [eds:Moreno-Diaz R. and Mira-Mira J., pp. 350-359.], MIT Press, Cambridge,MA,USA.
  • McCulloch WS and Pitts W (1943): A Logical Calculus Immanent in Nervous Activity. Bull.Math.Biophys. 5:115-133.

Crossrefs

Cf. A000609, A065246, A065248, A064436.

Formula

a(n)=A057156(n)-A000609(n)^n=A057156(n)-A065246(n).

A092258 (2^n-1)^(2^n).

Original entry on oeis.org

0, 1, 81, 5764801, 6568408355712890625, 529144398052420314716929933900838757437386767361
Offset: 0

Views

Author

N. J. A. Sloane, Feb 19 2004

Keywords

Comments

Number of maps from {0,1}^n to itself with no roots.

References

  • F. Robert, Discrete Iterations, Springer-Verlag, 1986, p. 167.

Crossrefs

Cf. A057156, A091324.

Programs

  • Mathematica
    (#-1)^#&/@(2^Range[0,5]) (* Harvey P. Dale, Feb 18 2024 *)

A340667 a(n) is derived from n by replacing each 0 in its binary representation with a string of n 0's.

Original entry on oeis.org

0, 1, 4, 3, 256, 65, 192, 7, 16777216, 524289, 2098176, 8195, 50331648, 49153, 114688, 15, 18446744073709551616, 4503599627370497, 36028797019226112, 1099511627779, 2305844108725321728, 17592190238721, 70368756760576, 67108871, 14167099448608935641088
Offset: 0

Views

Author

Alois P. Heinz, Jan 15 2021

Keywords

Links

Crossrefs

Main diagonal of A340666.
Cf. A000079, A000120, A000225, A023416, A057156.

Programs

  • Maple
    a:= n-> Bits[Join](subs(0=[0$n][], Bits[Split](n))):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, b(r, k)*2+1, b(r, k)*2^k))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
  • Python
    def A340667(n):
    return 0 if n == 0 else int(bin(n)[2:].replace('0','0'*n),2) # Chai Wah Wu, Jan 29 2021

Formula

a(n) = n <=> n in { A000225 }.
a(n) = n^n <=> n in { A000079 }.
A000120(a(n)) = A000120(n).
A023416(a(n)) = n * A023416(n) for n >= 1.

A091324 (2^n)^(2^n)-(2^n-1)^(2^n).

Original entry on oeis.org

1, 3, 175, 11012415, 11878335717996660991, 932357239278482603486754898815444262218545775615, 25020804906535555378545307822902185893828140712948517836661141440629855580651950069203769685588364042197142274707455
Offset: 0

Views

Author

N. J. A. Sloane, Feb 19 2004

Keywords

Comments

Number of maps from {0,1}^n to itself with at least one root.

References

  • F. Robert, Discrete Iterations, Springer-Verlag, 1986, p. 167.

Crossrefs

Cf. A057156, A092258.
Showing 1-8 of 8 results.

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

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