A199116 -id:A199116 - cp's OEIS frontend

A221130 a(n) = 2^(2*n - 1) + n.

3, 10, 35, 132, 517, 2054, 8199, 32776, 131081, 524298, 2097163, 8388620, 33554445, 134217742, 536870927, 2147483664, 8589934609, 34359738386, 137438953491, 549755813908, 2199023255573, 8796093022230, 35184372088855, 140737488355352, 562949953421337

Offset: 1

Jaroslav Krizek, Jan 02 2013

nonn

Conjecture: a(n ) = the smallest numbers w such that numbers w, w+1,…, w+k-1 for k=1,2,…n are numbers of form h*2^m + m, where 1<=h <2^m, m = natural number (see A221129).
a(5) = 517 because numbers 517, 518, 519, 520, 521 are numbers of presented form.
517 = 16*2^5 + 5, 518 = 8*2^6 + 6, 519 = 4*2^7 + 7, 520 = 2*2^8 + 8, 521 = 1*2^9 + 9 (that is, numbers (2^(n-k))*(2^(n+k-1))+n+k-1, for k=1,2,,...n).

			a(5)=2^(2*5-1)+5=517.

Jaroslav Krizek, Table of n, a(n) for n = 1..53
Index entries for linear recurrences with constant coefficients, signature (6,-9,4)

Cf. A221129, A199116.

a(n+1) = a(n) + 3*2^(2*n-1)+1 = a(n) + 6*4^(n-1)+1 = a(n) + 2^(2*n+1) - 2^(2*n-1) + 1 = a(n) + A199116(n-1).

G.f. -x*(3-8*x+2*x^2) / ( (4*x-1)*(x-1)^2 ). - R. J. Mathar, Jan 17 2013

A221129 Numbers of form k*2^n + n where 1 <= k < 2^n and n > 0.

Original entry on oeis.org

3, 6, 10, 11, 14, 19, 20, 27, 35, 36, 37, 43, 51, 52, 59, 68, 69, 70, 84, 100, 101, 116, 132, 133, 134, 135, 148, 164, 165, 180, 196, 197, 198, 212, 228, 229, 244, 261, 262, 263, 264, 293, 325, 326, 357, 389, 390, 391, 421, 453, 454, 485, 517, 518, 519, 520, 521

Offset: 1

Jaroslav Krizek, Jan 02 2013

nonn

Subsequence of primes: 3, 11, 19, 37, 43, 59, 101, 197, 229, 263, ...

			a(6) = 19 = 2 *2^3 + 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A221130, A199116.

N:= 1000: # to get all terms <= N
sort([seq(seq(k*2^n+n,k=1..min(2^n-1,(N-n)/2^n)),n=1..ilog2(N))]); # Robert Israel, Jan 21 2019

seqUpTo[n_] := Select[Union@Flatten@Table[k*2^h + h, {h, Floor@Log[2, n]}, {k, Min[2^h - 1, Floor[n/2^h]]}], # <= n &]; seqUpTo[521] (* Giovanni Resta, Jan 08 2013 *)

A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

Original entry on oeis.org

1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886

Offset: 0

Paul Curtz, Dec 28 2016

a(n) mod 9 = period 2: repeat [1, 7].
The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
a(n) is a companion to A051049(n).
With an initial 0, A051049(n) is an autosequence of the first kind.
With an initial 2, this sequence is an autosequence of the second kind.
See the reference.
Difference table:
1,   7, 10, 25, 46,  97, ... = this sequence.
6,   3, 15, 21, 51,  93, ... = 3*A014551(n)
-3, 12,  6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...

			a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.

seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # Peter Luschny, Dec 29 2016

LinearRecurrence[{2,1,-2},{1,7,10},50] (* Paolo Xausa, Nov 13 2023 *)

Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016

a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
a(n) = 2*A051049(n+1) - A051049(n).
From Colin Barker, Dec 28 2016: (Start)
a(n) = 3*2^n - 2 for n even.
a(n) = 3*2^n + 1 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Original entry on oeis.org

3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888

Offset: 0

Paul Curtz, Jan 01 2017

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
Difference table of a(n):
3,   7, 12, 25, 48,  97, 192, ...
4,   5, 13, 23, 49,  95, 193, ...  = -(-1)^n* A140683(n)
1,   8, 10, 26, 46,  98, 190, ...  = A259713(n)
7,   2, 16, 20, 52,  92, 196, ...
-5, 14,  4, 32, 40, 104, 184, ...
... .

			a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)

Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n+2) = a(n) + 9*2^n.
a(n) = 2^(n+2) - A051049(n).
From Colin Barker, Jan 01 2017: (Start)
a(n) = 3*2^n for n even.
a(n) = 3*2^n + 1 for n odd.
G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
Binomial transform of 3, followed by (-1)^n* A140657(n).

More terms from Colin Barker, Jan 01 2017

cp's OEIS Frontend

A221130 a(n) = 2^(2*n - 1) + n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A221129 Numbers of form k*2^n + n where 1 <= k < 2^n and n > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions