A265736 -id:A265736 - cp's OEIS frontend

A020522 a(n) = 4^n - 2^n.

0, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552, 4192256, 16773120, 67100672, 268419072, 1073709056, 4294901760, 17179738112, 68719214592, 274877382656, 1099510579200, 4398044413952, 17592181850112, 70368735789056, 281474959933440

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of walks of length 2*n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba, Jul 02 2004
If we consider a(4*k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4*k+2) + 3^(4*k+2) == 3^k*(4+9) == 3*0 == 0 (mod 13). So a(4*k+2) can never be prime. - Jose Brox, Dec 27 2005
If k is odd, then a(n*k) is divisible by a(n), since: a(n*k) = (2^n)^k + (3^n)^k = (2^n + 3^n)*((2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)). So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005
Let x,y,z be elements from some power set P(n), i.e., the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye, Dec 26 2005
Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev, Jan 13 2006
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye, Jan 02 2008
For n>1: central terms of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010
Pronic numbers of the form: (2^n - 1)*2^n, which is the n-th Mersenne number times 2^n, see A000225 and A002378. - Fred Daniel Kline, Nov 30 2013
Indices where records of A037870 occur. - Philippe Beaudoin, Sep 03 2014
Half the total edge length for a minimum linear arrangement of a hypercube of dimension n. (See Harper's paper below for proof). - Eitan Frachtenberg, Apr 07 2017
Number of pairs in GF(2)^{n+1} whose dot product is 1. - Christopher Purcell, Dec 11 2021

			n=5: a(5) = 4^5 - 2^5 = 1024 - 32 = 992 -> '1111100000'.

Vincenzo Librandi, Table of n, a(n) for n = 0..170
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Tom Copeland, The Kervaire-Milnor formula
John Elias, Illustration of initial terms: Twin 2^n hexagonal numbers
L. H. Harper, Optimal Assignment of Numbers to Vertices, J. SIAM 12(1), p. 131--135, March 1964; alternative link.
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.
The Sixtieth William Lowell Putnam Mathematical Competition, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Ratio of successive terms of A028365.
Cf. A000225, A060867, A161168, A006516, A059153, A065442.
Cf. A000079, A265736, A020988, A047849.
Cf. A000079, A000302.

a020522 = (* 2) . a006516  -- Reinhard Zumkeller, Dec 15 2015

[4^n - 2^n: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

A020522:=n->4^n-2^n; seq(A020522(n), n=0..50); # Wesley Ivan Hurt, Nov 29 2013

Table[4^n - 2^n, {n, 40}] (* or *) LinearRecurrence[{6, -8}, {0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n)=4^n-2^n \\ Charles R Greathouse IV, Jan 30 2012

def A020522(n): return (1<Chai Wah Wu, Mar 10 2025

[4^n - 2^n for n in range(0,23)] # Zerinvary Lajos, Jun 05 2009

From Herbert Kociemba, Jul 02 2004: (Start)
G.f.: 2*x/((-1 + 2*x)*(-1 + 4*x)).
a(n) = 6*a(n-1) - 8*a(n-2). (End)
E.g.f.: exp(4*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009
From Reinhard Zumkeller, Feb 07 2006, Jaroslav Krizek, Aug 02 2009: (Start)
a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n).
In binary representation, n>0: n 1's followed by n 0's (A138147(n)).
A000120(a(n)) = n.
A023416(a(n)) = n.
A070939(a(n)) = 2*n.
2*a(n)+1 = A030101(A099393(n)). (End)
a(n) = A085812(n) - A001700(n). - John Molokach, Sep 28 2013
a(n) = 2*A006516(n) = A000079(n)*A000225(n) = A265736(A000225(n)). - Reinhard Zumkeller, Dec 15 2015
a(n) = (4^(n/2) - 4^(n/4))*(4^(n/2) + 4^(n/4)). - Bruno Berselli, Apr 09 2018
Sum_{n>0} 1/a(n) = E - 1, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A000302(n) - A000079(n). - John Reimer Morales, Aug 04 2025

A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 3, 3, 3, 7, 6, 5, 4, 7, 7, 7, 5, 5, 7, 7, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 15, 14, 13, 12, 11, 10, 9, 8, 15, 15, 15, 13, 13, 11, 11, 9, 9, 15, 15, 15, 14, 15, 14, 11, 10, 11, 10, 15, 14, 15, 15, 15, 15, 15, 11, 11, 11, 11, 15

Offset: 0

Reinhard Zumkeller, Dec 15 2015

			.          10 | 1010                            12 | 1100
.           4 |  100                             6 |  110
.   ----------+-----                     ----------+-----
.   4 IMPL 10 | 1011 -> T(10,4)=11       6 IMPL 12 | 1101 -> T(12,6)=13
.
First 16 rows of the triangle, where non-symmetrical rows are marked, see comment concerning A158582 and A089633:
.   0:                                 0
.   1:                               1   1
.   2:                             3   2   3
.   3:                           3   3   3   3
.   4:                         7   6   5   4   7    X
.   5:                       7   7   5   5   7   7
.   6:                     7   6   7   6   7   6   7
.   7:                   7   7   7   7   7   7   7   7
.   8:                15  14  13  12  11  10   9   8  15    X
.   9:              15  15  13  13  11  11   9   9  15  15    X
.  10:            15  14  15  14  11  10  11  10  15  14  15    X
.  11:          15  15  15  15  11  11  11  11  15  15  15  15
.  12:        15  14  13  12  15  14  13  12  15  14  13  12  15    X
.  13:      15  15  13  13  15  15  13  13  15  15  13  13  15  15
.  14:    15  14  15  14  15  14  15  14  15  14  15  14  15  14  15
.  15:  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15 .

Reinhard Zumkeller, Rows n = 0..255 of triangle, flattened
Eric Weisstein's World of Mathematics, Implies

Cf. A003817, A007088, A029578, A089633, A158582, A247648, A265716 (central terms), A265736 (row sums).
Cf. A053644, A265885, A327490.
Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A102037 (CNIMPL).

a265705_tabl = map a265705_row [0..]
a265705_row n = map (a265705 n) [0..n]
a265705 n k = k `bimpl` n where
   bimpl 0 0 = 0
   bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
               where (p', u) = divMod p 2; (q', v) = divMod q 2

using IntegerSequences
for n in 0:15 println(n == 0 ? [0] : [Bits("IMP", k, n) for k in 0:n]) end  # Peter Luschny, Sep 25 2021

A265705 := (n, k) -> Bits:-Implies(k, n):
seq(seq(A265705(n, k), k=0..n), n=0..11); # Peter Luschny, Sep 23 2019

T[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[n, 2]]-1-k, n]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's PARI code *)

T(n, k) = if(n==0,return(0)); bitor((2<David A. Corneth, Sep 24 2021

T(n,0) = T(n,n) = A003817(n).
T(2*n,n) = A265716(n).
Let m = A089633(n): T(m,k) = T(m,m-k), k = 0..m.
Let m = A158582(n): T(m,k) != T(m,m-k) for at least one k <= n.
Let m = A247648(n): T(2*m,m) = 2*m.
For n > 0: A029578(n+2) = number of odd terms in row n; no even terms in odd-indexed rows.
A265885(n) = T(prime(n),n).
A053644(n) = smallest k such that row k contains n.

A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

Original entry on oeis.org

0, 1, 5, 6, 22, 23, 27, 28, 92, 93, 97, 98, 114, 115, 119, 120, 376, 377, 381, 382, 398, 399, 403, 404, 468, 469, 473, 474, 490, 491, 495, 496, 1520, 1521, 1525, 1526, 1542, 1543, 1547, 1548, 1612, 1613, 1617, 1618, 1634, 1635, 1639, 1640, 1896, 1897, 1901, 1902, 1918

Offset: 0

Alex Ratushnyak, Apr 19 2013

			a(2) = (0 xor 2) + (1 xor 2) = 2 + 3 = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A001196 (bit doubling).
Row sums of A051933.
Other sums: A222423 (AND), A350093 (OR), A265736 (IMPL), A350094 (CNIMPL), A004125 (mod).

read("transforms"):
A051933 := proc(n,k)
    XORnos(n,k) ;
end proc:
A224915 := proc(n)
    add(A051933(n,k),k=0..n) ;
end proc: # R. J. Mathar, Apr 26 2013
# second Maple program:
with(MmaTranslator[Mma]):
seq(add(BitXor(n,i),i=0..n),n=0..60); # Ridouane Oudra, Dec 09 2020

Array[Sum[BitXor[#, k], {k, 0, #}] &, 53, 0] (* Michael De Vlieger, Dec 09 2020 *)

a(n) = sum(k=0, n, bitxor(n, k)); \\ Michel Marcus, Jun 08 2019

a(n) = (3*fromdigits(binary(n),4) - n) >>1; \\ Kevin Ryde, Dec 17 2021

for n in range(59):
    s = 0
    for k in range(n):  s += n ^ k
    print(s, end=',')

def A224915(n): return 3*int(bin(n)[2:],4)-n>>1 # Chai Wah Wu, Aug 21 2023

a(n) = Sum_{j=1..n} 4^(v_2(j)), where v_2(j) is the exponent of highest power of 2 dividing j. - Ridouane Oudra, Jun 08 2019
a(n) = n + 3*Sum_{j=1..floor(log_2(n))} 4^(j-1)*floor(n/2^j), for n>=1. - Ridouane Oudra, Dec 09 2020
From Kevin Ryde, Dec 17 2021: (Start)
a(2*n+b) = 4*a(n) + n + b where b = 0 or 1.
a(n) = (A001196(n) - n)/2.
a(n) = A350093(n) - A222423(n), being XOR = OR - AND.
(End)

A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

Original entry on oeis.org

0, 2, 7, 12, 26, 34, 45, 56, 100, 114, 131, 148, 174, 194, 217, 240, 392, 418, 447, 476, 514, 546, 581, 616, 684, 722, 763, 804, 854, 898, 945, 992, 1552, 1602, 1655, 1708, 1770, 1826, 1885, 1944, 2036, 2098, 2163, 2228, 2302, 2370, 2441, 2512, 2712, 2786, 2863

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of n OR k is to force a 1-bit at all bit positions where n has a 1-bit, which means n*(n+1) in the sum. Bits of k where n has a 0-bit are NOT(n) AND k = n CNIMPL k so that a(n) = A350094(n) + n*(n+1).

Winston de Greef, Table of n, a(n) for n = 0..10000

Cf. A003986 (bitwise OR), A001196 (bit doubling).
Row sums of A080098.
Other sums: A222423 (AND), A224915 (XOR), A265736 (IMPL), A350094 (CNIMPL).

a(n) = (3*(n^2 + fromdigits(binary(n),4)) + 2*n) >> 2;

a(n) = ((3*n+2)*n + A001196(n)) / 4.
a(2*n) = 4*a(n) - n.
a(2*n+1) = 4*a(n) + 2*n + 2.
a(n) = A222423(n) + A224915(n), being OR = AND + XOR.

A350094 a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).

Original entry on oeis.org

0, 0, 1, 0, 6, 4, 3, 0, 28, 24, 21, 16, 18, 12, 7, 0, 120, 112, 105, 96, 94, 84, 75, 64, 84, 72, 61, 48, 42, 28, 15, 0, 496, 480, 465, 448, 438, 420, 403, 384, 396, 376, 357, 336, 322, 300, 279, 256, 360, 336, 313, 288, 270, 244, 219, 192, 196, 168, 141, 112

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of NOT(n) AND k is to retain from k only those bits where n has a 0-bit. Conversely n AND k retains from k those bits where n has a 1-bit. Together they are all bits of k so that a(n) + A222423(n) = Sum_{k=0..n} k = n*(n+1)/2.

Row sums of A102037.
Cf. A001196 (bit doubling).
Other sums: A222423 (AND), A350093 (OR), A224915 (XOR), A265736 (IMPL).

with(Bits): cnimp := (n, k) -> And(Not(n), k):
seq(add(cnimp(n, k), k = 0..n), n = 0..59); # Peter Luschny, Dec 14 2021

a(n) = (3*fromdigits(binary(n),4) - n^2 - 2*n)/4;

a(n) = (A001196(n) - n*(n+2))/4.
a(2*n) = 4*a(n) + n.
a(2*n+1) = 4*a(n).

cp's OEIS Frontend

A020522 a(n) = 4^n - 2^n.

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A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

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A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

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A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

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A350094 a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).

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Programs

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