A049775 -id:A049775 - cp's OEIS frontend

A007582 a(n) = 2^(n-1)*(1+2^n).

1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136, 140737496743936, 562949970198528

Offset: 0

Simon Plouffe

Let G_n be the elementary Abelian group G_n = (C_2)^n for n >= 1: A006516 is the number of times the number -1 appears in the character table of G_n and A007582 is the number of times the number 1. Together the two sequences cover all the values in the table, i.e., A006516(n) + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 01 2001
Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_8. Example: a(1)=3 because in the cycle ABCDEFGH we have three walks of length 3 between A and B: ABAB, ABCB and AHAB. - Emeric Deutsch, Apr 01 2004
Smallest number containing in its binary representation two equal non-overlapping subwords of length n: A097295(a(n))=n and A097295(m)Reinhard Zumkeller, Aug 04 2004
a(n)^2 + (A006516(n))^2 = a(2n). E.g., a(3) = 36, A006516(3) = 28, a(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson, Jun 17 2006
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either x equals y or x does not equal y. - Ross La Haye, Jan 02 2008
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A). This is just a simpler statement of my previous comment for this sequence. - Ross La Haye, Jan 10 2008
For n>0: A000120(a(n))=2, A023414(a(n))=2*(n-1), A087117(a(n))=n-1. - Reinhard Zumkeller, Jun 23 2009
a(n+1) written in base 2: 11, 1010, 100100, 10001000, 1000010000, ..., i.e., number 1, n times 0, number 1, n times 0 (A163449(n)). - Jaroslav Krizek, Jul 27 2009
a(n) for n >= 1 is a bisection of A001445(n+1). - Jaroslav Krizek, Aug 14 2009
Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum_{k=0..n} binomial(n,k)*k^q, then A007582(x)= sum_{k=0..x-1} T(x,k)*2^k. - John M. Campbell, Nov 16 2011
a(n) gives the number of pairs (r, s) such that 0 <= r <= s <= (2^n)-1 that satisfy AND(r, s, XOR(r, s)) = 0. - Ramasamy Chandramouli, Aug 30 2012
a(n) = A000217(2^n) = 2^(2n-1) + 2^(n-1) is the nearest triangular number above 2^(2n-1); cf. A006516, A233327. - Antti Karttunen, Feb 26 2014
Consider the quantum spin-1/2 chain with even number of sites L (physics, condensed matter theory). The spectrum of the Hamiltonian can be classified according to symmetries. If the only symmetry of the spin Hamiltonian is Parity, i.e., reflection with respect to the middle of the chain (see e.g. the transverse-field Ising model with open boundary conditions), then the dimension of the p=+1 parity sector is given by a(n) with n=L/2. - Marin Bukov, Mar 11 2016
a(n) is also the total number of words of length n, over an alphabet of four letters, of which one of them appears an even number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter odd case), and the Balakrishnan reference there. For the 1- to 11-letter cases, see the crossrefs. - Wolfdieter Lang, Jul 17 2017
a(n) is the number of nonisomorphic spanning trees of the cyclic snake formed with n+1 copies of the cycle on 4 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m. - Christian Barrientos, Sep 05 2024
Also, with offset 1, the cogrowth sequence of the dihedral group with 16 elements, D8 = . - Sean A. Irvine, Nov 06 2024

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

T. D. Noe, Table of n, a(n) for n=0..200
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
S. Hong and J. H. Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 168
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Mircea Merca, A Note on Cosine Power Sums, J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Index entries for linear recurrences with constant coefficients, signature (6,-8)

Cf. A000217, A049773, A049775.
Cf. A006516.
Cf. A134308.
Cf. A000225, A000392, A032263, A028243, A000079.
Cf. A102573.
The number of words of length n with m letters, one of them appearing an even number of times is for m = 1..11: A000035, A011782,  A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192. - Wolfdieter Lang, Jul 17 2017

[Binomial(2^n + 1, 2) : n in [0..30]]; // Wesley Ivan Hurt, Jul 03 2020

seq(binomial(-2^n, 2), n=0..23); # Zerinvary Lajos, Feb 22 2008

Table[ Binomial[2^n + 1, 2], {n, 0, 23}] (* Robert G. Wilson v, Jul 30 2004 *)
LinearRecurrence[{6,-8},{1,3},30] (* Harvey P. Dale, Apr 08 2013 *)

A007582(n):=2^(n-1)*(1+2^n)$ makelist(A007582(n),n,0,30); /* Martin Ettl, Nov 15 2012 */

PARI
```
a(n)=if(n<0,0,2^(n-1)*(1+2^n))
```

a(n)=sum(k=-n\4,n\4,binomial(2*n+1,n+1+4*k))

G.f.: (1-3*x)/((1-2*x)*(1-4*x)). C(1+2^n, 2) where C(n, 2) is n-th triangular number A000217.
Binomial transform of A007051. Inverse binomial transform of A081186. - Paul Barry, Apr 07 2003
E.g.f.: exp(3*x)*cosh(x). - Paul Barry, Apr 07 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*3^(n-2*k). - Paul Barry, May 08 2003
a(n+1) = 4*a(n) - 2^n; see also A049775. a(n) = 2^(n-1)*A000051(n). - Philippe Deléham, Feb 20 2004
a(n) = 6*a(n-1) - 8*a(n-2). - Emeric Deutsch, Apr 01 2004
Row sums of triangle A134308. - Gary W. Adamson, Oct 19 2007
a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Mar 01 2008
a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Apr 02 2008
a(n) = A000079(n) + A006516(n). - Yosu Yurramendi, Aug 06 2008
a(n) = A028403(n+1) / 4. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=-floor(n/4)..floor(n/4)} binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012
G.f.: Q(0)/2 where Q(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
a(n) = Sum_{k=1..2^n} k. - Joerg Arndt, Sep 01 2013
a(n) = (1/3) * Sum_{k=2^n..2^(n+1)} k. - J. M. Bergot, Jan 26 2015
a(n+1) = 2*a(n) + 4^n. - Yuchun Ji, Mar 10 2017

A293697 a(n) is the sum of prime numbers between 2^n+1 and 2^(n+1).

Original entry on oeis.org

2, 3, 12, 24, 119, 341, 1219, 4361, 16467, 57641, 208987, 780915, 2838550, 10676000, 39472122, 148231324, 559305605, 2106222351, 7995067942, 30299372141, 115430379568, 440354051430, 1683364991290, 6448757014608, 24754017328490, 95132828618112, 366232755206338

Offset: 0

Olivier Gérard, Oct 15 2017

nonn

			From _Gus Wiseman_, Jun 02 2024: (Start)
Row-sums of:
   2
   3
   5   7
  11  13
  17  19  23  29  31
  37  41  43  47  53  59  61
  67  71  73  79  83  89  97 101 103 107 109 113 127
(End)

Cf. A036378 (number of primes summed).
Cf. A293696 (triangle of partial sums).
Minimum is A014210 or A104080, indices A372684.
Maximum is A014234, delta A013603.
Counting all numbers (not just prime) gives A049775.
For squarefree instead of prime numbers we have A373123, length A077643.
For prime indices we have A373124.
Partial sums give A130739(n+1).
Cf. A000040, A001223, A029931, A035100, A046933, A061398, A092131.

Table[Plus @@
  Table[Prime[i], {i, PrimePi[2^(n)] + 1, PrimePi[2^(n + 1)]}], {n, 0,
   24}]

A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

Original entry on oeis.org

1, 4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473, 6442450945, 12884901889

Offset: 0

N. J. A. Sloane, R. K. Guy

Also Pisot sequence L(4,7) (cf. A008776).
Alternatively, define the sequence S(a(1),a(2)) by: a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n > 0. This is S(4,7).
a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. E.g., a(2)=7 corresponds to (8,12,16,20,24,28,32). - Augustine O. Munagi, Feb 21 2007
Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3, ...]. - Gary W. Adamson, Aug 27 2010

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Sequences, 11 (2008), #08.5.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3, -2).
Index entries for Pisot sequences

Cf. A049988, A049775, A000051, A008776.
A181565 is an essentially identical sequence.
For primes see A002253 and A039687.

[1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015

A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation

s=4;lst={1,s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)
Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *)
{1}~Join~LinearRecurrence[{3, -2}, {4, 7}, 33] (* Michael De Vlieger, Dec 16 2015 *)

a(n)=3<Charles R Greathouse IV, Sep 28 2015

a(n) = 3a(n-1) - 2a(n-2).
For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - Benoit Cloitre, Aug 14 2002
For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - Philippe Deléham, Feb 20 2004
O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)). - R. J. Mathar, Nov 23 2007
For n>0, a(n) = 2*a(n-1)-1. - Vincenzo Librandi, Dec 16 2015
E.g.f.: exp(x)*(1 + 3*sinh(x)). - Stefano Spezia, May 06 2023

Edited by N. J. A. Sloane, Dec 16 2015 at the suggestion of Bruno Berselli

A049773 Triangular array T read by rows: if row n is r(1),...,r(m), then row n+1 is 2r(1)-1,...,2r(m)-1,2r(1),...,2r(m).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8, 16, 1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 2, 18, 10, 26, 6, 22, 14, 30, 4, 20, 12, 28, 8, 24, 16, 32, 1, 33, 17, 49, 9, 41, 25, 57, 5, 37, 21, 53, 13, 45, 29, 61, 3, 35, 19

Offset: 1

Clark Kimberling

n-th row = (r(1),r(2),...,r(m)), where m=2^(n-1), satisfies r(r(k))=k for k=1,2,...,m and has exactly 2^[ n/2 ] solutions of r(k)=k. (The function r(k) reverses bits. Or rather, r(k)=revbits(k-1)+1.)
In a knockout competition with m players, arranging the competition brackets (see links) in r(k) order, where k is the rank of the player, ensures that highest ranked players cannot meet until the later stages of the competition.  None of the top 2^p ranked players can meet earlier than the p-th from last round of the competition.  At the same time the top ranked players in each match meet the highest ranked player possible consistent with this rule.  The sequence for the top ranked players meeting the lowest ranked player possible is A131271. - Colin Hall, Jul 31 2011, Feb 29 2012
Row n contains one of A003407(2^(n-1)) non-averaging permutations of [2^(n-1)], i.e., a permutation of [2^(n-1)] without 3-term arithmetic progressions. - Alois P. Heinz, Dec 05 2017

			Triangle begins:
1;
1,  2;
1,  3, 2,  4;
1,  5, 3,  7, 2,  6,  4,  8;
1,  9, 5, 13, 3, 11,  7, 15, 2, 10,  6, 14, 4, 12,  8, 16;
1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 2, 18, 10, 26, ...

Alois P. Heinz, Rows n = 1..13, flattened
Eric Weisstein's World of Mathematics, Nonaveraging Sequence.
Wikipedia, Arithmetic progression.
Wikipedia, Bracket (tournament).
Index entries related to non-averaging sequences.

Sum of odd-indexed terms of n-th row gives A007582. Sum of even-indexed terms gives A049775.
A030109 is another version.
Cf. A131271.
Cf. A088370.
Cf. A003407, A088208.

a049773 n k = a049773_tabf !! (n-1) !! (k-1)
a049773_row n = a049773_tabf !! (n-1)
a049773_tabf = iterate f [1] where
   f vs = (map (subtract 1) ws) ++ ws where ws = map (* 2) vs
-- Reinhard Zumkeller, Mar 14 2015

T:= proc(n) option remember; `if`(n=1, 1,
      [map(x->2*x-1, [T(n-1)])[], map(x->2*x, [T(n-1)])[]][])
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, Oct 28 2011

row[1] = {1}; row[n_] := row[n] = Join[ 2*row[n-1] - 1, 2*row[n-1] ]; Flatten[ Table[ row[n], {n, 1, 7}]] (* Jean-François Alcover, May 03 2012 *)

(a(n, k) = if( k<=0 || k>=n, 0, if( k%2, n\2) + a(n\2, k\2))); {T(n, k) = if( k<=0 || k>2^n/2, 0, 1 + a(2^n/2, k-1))}; /* Michael Somos, Oct 13 1999 */

A171473 a(n) = 6a(n-1) - 8a(n-2)-3 for n > 1; a(0) = 35, a(1) = 135.

Original entry on oeis.org

35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135

Offset: 0

Klaus Brockhaus, Dec 09 2009

Related to Reverse and Add trajectory of 22 in base 2: A061561(4*n+3) = 3*a(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A061561, A092431, A049775, A171470, A171471, A171472.

[32*4^n+4*2^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

{m=20; v=concat([35, 135], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]-3); v}

a(n) = 32*4^n + 4*2^n - 1.
G.f.: (35-110*x+72*x^2)/((1-x)*(1-2*x)*(1-4*x)).
a(n) = A092431(n+3).
a(n+1) - a(n) = A049775(n+5).
E.g.f.: exp(x)*(32*exp(3*x) + 4*exp(x) - 1). - Stefano Spezia, Sep 27 2023

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

Original entry on oeis.org

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset: 1

Reinhard Zumkeller, Mar 23 2004

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007088, A001700.
Subsequence of A031443.
Cf. A000120, A023416.
Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},n-2,1]],2],{n,2,30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)),{x,0,30}],x]] (* Harvey P. Dale, Jul 30 2021 *)

a(n+1) = 2*a(n) + 4^n + 1.
a(n) = 2^(2*n-1) + 2^(n-1) - 1.
a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004
G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

A110613 a(n+3) = 5a(n+2) - 2a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.

Original entry on oeis.org

1, 0, 3, 7, 29, 107, 421, 1659, 6597, 26299, 105029, 419771, 1678405, 6712251, 26846277, 107379643, 429507653, 1718008763, 6871991365, 27487878075, 109951337541, 439805000635, 1759219303493, 7036875815867, 28147500467269

Offset: 0

Creighton Dement, Jul 31 2005

A Jacobsthal related sequence (A001045). This sequence was calculated using the same rules given for A108618; the "initial seed" is the floretion given in the program code, below.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-2,-8).

Cf. A001045, A007582, A007581, A049775, A087440, A108618, A110614.

seriestolist(series((1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2tessumseq[(.5'i - .5'k - .5i' + .5k' - .5'ij' - .5'ji' - .5'jk' - .5'kj')('i + j' + 'ij' + 'ji')] Sumtype is set to:sum[Y[15]] = sum(*) (from 3rd term, disregarding signs)

LinearRecurrence[{5, -2, -8}, {1, 0, 3}, 50] (* G. C. Greubel, Sep 01 2017 *)

x='x+O('x^50); Vec((1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1))) \\ G. C. Greubel, Sep 01 2017

G.f.: (1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1)).
Program "Superseeker" finds:
a(n) + a(n+1) = A007582(n) = A007581(n+1) - A007581(n).
a(n+2) - a(n) = A049775(n).
a(n) + 2*a(n+1) + a(n+2) = A087440(n+1).

A294193 a(n) = sum of integers between n!+1 and (n+1)!.

Original entry on oeis.org

0, 2, 18, 279, 6960, 252300, 12443760, 800168040, 65028257280, 6518255405760, 790091384544000, 113924591159702400, 19273172758289049600, 3780639334294658035200, 851206099134433961318400, 218026562222345234117760000, 63037891684425054948655104000

Offset: 0

Olivier Gérard, Oct 24 2017

Useful as a growth reference for sequences summing on intervals between 2 factorials.

			a(2) = 3 + 4 + 5 + 6 = 18.
a(3) = 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 = 24*25/2 - 6*7/2 = 279.

Iain Fox, Table of n, a(n) for n = 0..252

Cf. A000217 (triangular numbers).
Cf. A001563 (difference of factorials).
Cf. A049775 (same idea between consecutive powers of 2).
Cf. A000142, A055555.

Table[1/2 ((n + 1)! ((n + 1)! + 1) - n! (n! + 1) ), {n, 0, 10}]

a(n) = 1/2*((n+1)!*((n+1)! + 1)-n!*(n!+1)) \\ Iain Fox, Nov 28 2017

a(n) = (1/2) * ((n + 1)!*((n + 1)! + 1) - n!*(n! + 1) ).

a(n) = A055555(n+1) - A055555(n). - J.S. Seneschal, Jul 07 2025

More terms from Iain Fox, Nov 28 2017

cp's OEIS Frontend

A007582 a(n) = 2^(n-1)*(1+2^n).

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A293697 a(n) is the sum of prime numbers between 2^n+1 and 2^(n+1).

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A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

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A049773 Triangular array T read by rows: if row n is r(1),...,r(m), then row n+1 is 2r(1)-1,...,2r(m)-1,2r(1),...,2r(m).

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A171473 a(n) = 6*a(n-1) - 8*a(n-2)-3 for n > 1; a(0) = 35, a(1) = 135.

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Magma

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A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

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A110613 a(n+3) = 5*a(n+2) - 2*a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.

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A171473 a(n) = 6a(n-1) - 8a(n-2)-3 for n > 1; a(0) = 35, a(1) = 135.

A110613 a(n+3) = 5a(n+2) - 2a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.

A158561 a(n) = ((2^n)((2^n)+1) - (2^(n-1))((2^(n-1))+1))/2, a(1)=3.