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 1-4 of 4 results.

A000918 a(n) = 2^n - 2.

Original entry on oeis.org

-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 68719476734, 137438953470
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011
For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002
For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004
Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004
For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005
Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005
From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005
The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006
Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007
Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007
It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009
First differences of A005803. - Reinhard Zumkeller, Oct 12 2011
For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014
For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015
Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015
Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020
For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019
The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020
For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020
For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022
For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

Examples

			a(4) = 14 because the 14 = 6 + 4 + 4 rationals (in lowest terms) for n = 3 are (see the Jun 14 2017 formula above): 1/2, 1, 3/2, 2, 5/2, 3; 1/4, 3/4, 5/4, 7/4; 1/8, 3/8, 5/8, 7/8. - _Wolfdieter Lang_, Jun 14 2017

References

  • H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
  • Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. - Mohammad K. Azarian, Oct 27 2011
  • S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 86, Exercise 3.16.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
  • 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).
  • A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Links

Crossrefs

Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121, A110146.
Cf. A033484, A000225, A000325, A095151, A125128.
Cf. A058809 (3^n-3, n>0).

Programs

  • Haskell
    a000918 = (subtract 2) . (2 ^)
a000918_list = iterate ((subtract 2) . (* 2) . (+ 2)) (- 1)
-- Reinhard Zumkeller, Apr 23 2013
  • Magma
    [2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011
  • Maple
    seq(2^n-2,n=0..20) ;
  • Mathematica
    Table[2^n - 2, {n, 0, 29}] (* Alonso del Arte, Dec 01 2012 *)
  • PARI
    a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011
  • Python
    def A000918(n): return (1<Chai Wah Wu, Jun 10 2025

Formula

a(n) = 2*A000225(n-1).
G.f.: 1/(1 - 2*x) - 2/(1 - x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
For n >= 1, a(n) = A008970(n + 1, 2). - Philippe Deléham, Feb 21 2004
G.f.: (3*x - 1)/((2*x - 1)*(x - 1)). - Simon Plouffe in his 1992 dissertation for the sequence without the leading -1
a(n) = 2*a(n - 1) + 2. - Alexandre Wajnberg, Apr 25 2005
a(n) = A000079(n) - 2. - Omar E. Pol, Dec 16 2008
a(n) = A058896(n)/A052548(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = A164874(n - 1, n - 1) for n > 1. - Reinhard Zumkeller, Aug 29 2009
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n > 0. - Reinhard Zumkeller, Feb 28 2010
a(n + 1) = A027383(2*n - 1). - Jason Kimberley, Nov 02 2011
G.f.: U(0) - 1, where U(k) = 1 + x/(2^k + 2^k/(x - 1 - x^2*2^(k + 1)/(x*2^(k + 1) - (k + 1)/U(k + 1) ))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n+1) is the sum of row n in triangle A051601. - Reinhard Zumkeller, Aug 05 2013
a(n+1) = A127330(n,0). - Reinhard Zumkeller, Nov 16 2013
a(n) = Sum_{k=1..n-1} binomial(n, k) for n > 0. - Dan McCandless, Nov 14 2015
From Miquel Cerda, Aug 16 2016: (Start)
a(n) = A000225(n) - 1.
a(n) = A125128(n-1) - A000325(n).
a(n) = A095151(n) - A125128(n) - 1. (End)
a(n+1) = 2*(n + Sum_{j=1..n-1} (n-j)*2^(j-1)), n >= 1. This is the number of the rationals k/2, k = 1..2*n for n >= 1 and (2*k+1)/2^j for j = 2..n, n >= 2, and 2*k+1 < n-(j-1). See the example for n = 3 below. Motivated by the proposal A287012 by Mark Rickert. - Wolfdieter Lang, Jun 14 2017

Extensions

Maple programs fixed by Vaclav Kotesovec, Dec 13 2014

A145071 Partial sums of A000051, starting at n=1.

Original entry on oeis.org

3, 8, 17, 34, 67, 132, 261, 518, 1031, 2056, 4105, 8202, 16395, 32780, 65549, 131086, 262159, 524304, 1048593, 2097170, 4194323, 8388628, 16777237, 33554454, 67108887, 134217752, 268435481, 536870938, 1073741851, 2147483676, 4294967325, 8589934622, 17179869215
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

Keywords

Comments

The third number that is a sum of n positive n-th powers. - Alois P. Heinz, Aug 02 2020

Examples

			a(2) = a(1) + 2^2 + 1 = 3 + 4 + 1 = 8; a(3) = a(2) + 2^3 + 1 = 8 + 8 + 1 = 17.

Links

Crossrefs

Cf. A000051 (2^n + 1), A000225 (2^n - 1), A000295 (Eulerian numbers).
Cf. A052944, A127330.
Column k = 1 of triangle A308737.
Row n=3 of A336725.

Programs

  • ARIBAS
    a:=0; for n:=1 to 30 do a:=a+2**n+1; write(a,","); end;
  • Haskell
    a145071 n = 2 ^ (n + 1) + n - 2
a145071_list = scanl1 (+) $ tail a000051_list
-- Reinhard Zumkeller, Nov 16 2013
  • Mathematica
    lst={};s=0;Do[s+=2^n+1;AppendTo[lst,s],{n,5!}];lst
Accumulate[2^Range[30]+1] (* Harvey P. Dale, Feb 19 2023 *)

Formula

a(1) = 3; a(n) = a(n-1) + 2^n + 1 for n > 1.
a(n) = 2^(n+1) + n - 2. - Franklin T. Adams-Watters, Jul 06 2009
G.f.: x*(3-4*x)/((1-x)^2*(1-2*x)). - Colin Barker, Jan 11 2012
a(n) = A127330(n,n) = A052944(n-1) + 2. - Reinhard Zumkeller, Nov 16 2013
From Elmo R. Oliveira, Apr 01 2025: (Start)
E.g.f.: exp(x)*(x - 2 + 2*exp(x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3. (End)

Extensions

Edited by Klaus Brockhaus, Oct 14 2008

A129143 Start with the empty sequence and append in step k the consecutive numbers 2^(k-1) to 2^(k-1)+k-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 16, 17, 18, 19, 20, 32, 33, 34, 35, 36, 37, 64, 65, 66, 67, 68, 69, 70, 128, 129, 130, 131, 132, 133, 134, 135, 256, 257, 258, 259, 260, 261, 262, 263, 264, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 1024, 1025, 1026, 1027, 1028
Offset: 1

Views

Author

Klaus Brockhaus, Mar 31 2007

Keywords

Comments

Inspired by A127330.

Examples

			In step 5 the numbers 2^4 to 2^4+4 are appended: 16, 17, 18, 19, 20.

Links

Crossrefs

Cf. A127330, A129142.

Programs

  • Magma
    &cat[ [2^k..2^k+k]: k in [0..11] ];
  • Mathematica
    Table[2^n + k, {n, 0, 10}, {k, 0, n}] // Flatten (* Ivan Neretin, Aug 30 2015 *)

A129142 Start with the empty sequence and append in step k the consecutive numbers 2^k-1 to 2^k+k-2.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 15, 16, 17, 18, 31, 32, 33, 34, 35, 63, 64, 65, 66, 67, 68, 127, 128, 129, 130, 131, 132, 133, 255, 256, 257, 258, 259, 260, 261, 262, 511, 512, 513, 514, 515, 516, 517, 518, 519, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 2047
Offset: 1

Views

Author

Klaus Brockhaus, Mar 31 2007

Keywords

Comments

Inspired by A127330.

Examples

			In step 4 the numbers 2^4-1 to 2^4+4-2 are appended: 15, 16, 17, 18.

Crossrefs

Cf. A127330, A129143.

Programs

  • Magma
    &cat[ [2^k-1..2^k+k-2]: k in [1..11] ];
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.