A098011 -id:A098011 - cp's OEIS frontend

A176414 Expansion of (7+8x)/(1+2x).

7, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368

Offset: 0

Klaus Brockhaus, Apr 17 2010

Inverse binomial transform of A176415.

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

Cf. A176415, A110164 (essentially the same), A122803.

Join[{7},NestList[-2#&,-6,40]] (* Harvey P. Dale, Jun 20 2020 *)

{for(n=0, 29, print1(polcoeff((7+8*x)/(1+2*x)+x*O(x^n), n), ", "))}

A176414(n)=3*(-2)^n+!n*4 \\ M. F. Hasler, Apr 19 2015

a(n) = A110164(n+2) for n > 0.
a(n) = 3*(-2)^n = 3*A122803(n+1) for n > 0; a(0) = 7.
a(n) = -2*a(n-1) for n > 1; a(0) = 7, a(1) = -6.
a(n) = (-1)^n*A132477(n) = (-1)^n*A122391(n+3), n>1.
a(n) = (-1)^n*A111286(n+2) = (-1)^n*A098011(n+4) = (-1)^n*A091629(n) = (-1)^n*A087009(n+3) = (-1)^n*A082505(n+1) = (-1)^n*A042950(n+1) = (-1)^n*A007283(n) = (-1)^n*A003945(n+1), n>0. - R. J. Mathar, Dec 10 2010
E.g.f.: 4 + 3*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited by M. F. Hasler, Apr 19 2015

A348331 Lexicographically earliest sequence of positive integers such that for any n > 0, the sum of the indices k < n such that a(k) = a(n) is less than or equal to n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 2, 4, 4, 1, 5, 5, 3, 6, 6, 2, 7, 7, 4, 8, 8, 1, 9, 9, 5, 10, 10, 3, 11, 11, 6, 12, 12, 2, 13, 13, 7, 14, 14, 4, 15, 15, 8, 16, 16, 1, 17, 17, 9, 18, 18, 5, 19, 19, 10, 20, 20, 3, 21, 21, 11, 22, 22, 6, 23, 23, 12, 24, 24, 2, 25, 25, 13

Offset: 1

Rémy Sigrist, Oct 13 2021

nonn

Every integer appears infinitely many times.

See table A348357 where the integers are arranged according to the present sequence.

Cf. A098011, A348330, A348356, A348357.

s=vector(25); for (n=1, 75, for (k=1, oo, if (s[k]<=n, print1 (k", "); s[k]+=n; break)))

a(n) = 1 iff n belongs to A098011.

A114958 a(n) = 62^(n+1) - 5(n+1) - 4.

Original entry on oeis.org

3, 10, 29, 72, 163, 350, 729, 1492, 3023, 6090, 12229, 24512, 49083, 98230, 196529, 393132, 786343, 1572770, 3145629, 6291352, 12582803, 25165710, 50331529, 100663172, 201326463, 402653050, 805306229, 1610612592, 3221225323

Offset: 0

Creighton Dement, Feb 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A098011, A042950, A110164, A101229, A058764, A087009, A111286, A007283.

[6*2^(n+1) - 5*(n+1) - 4: n in [0..30] ]; // Vincenzo Librandi, May 18 2011

Vec((3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Apr 30 2019

From Colin Barker, Apr 30 2019: (Start)
G.f.: (3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>2.
(End)

A266926 a(0)=0, a(1)=1, a(2)=10; for n>2, a(n) = concat(a(1), ..., a(n-1)).

Original entry on oeis.org

0, 1, 10, 110, 110110, 110110110110, 110110110110110110110110, 110110110110110110110110110110110110110110110110, 110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110

Offset: 0

Giovanni Teofilatto, Jan 06 2016

Decimal conversions: 0, 1, 2, 6, 54, 3510, 14380470, 241264265751990, 67909853583655146508751957430, ... . (See A267348.) - Michael De Vlieger, Jan 06 2016
After 10, a(n) is '110' repeated 2^(n-3) times. Therefore, for n>3, a(n) is the concatenation of a(n-1) with itself.
After 1, each term with the 0's omitted is a member of A136308.
The number of digits in a(n) is A098011(n+1).
The number of digits in a(n+2)/a(n+1) gives A103204 with 2 repeated.

			a(3) = concat(1, 10, 110) = 110110.
a(4) = concat(1, 10, 110, 110110) = 110110110110.

Cf. A000079, A098011, A103204, A136308, A267348.

[n le 2 select n*5^(n-1) else 110*(10^(3*2^(n-3))-1)/999: n in [0..8]]; // Bruno Berselli, Jan 29 2016

a = {0, 1}; Do[AppendTo[a, FromDigits@ Flatten@ Map[IntegerDigits@ # &, If[n < 2, Reverse@ a, a]]], {n, 8}]; a (* Michael De Vlieger, Jan 06 2016 *)

a(n) = 110*(10^(3*2^(n-3))-1)/999 for n>2. - Bruno Berselli, Jan 29 2016

Definition by Michael De Vlieger, Jan 06 2016

Edited by Editors of the OEIS, Jan 29 2016

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

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A176414 Expansion of (7+8*x)/(1+2*x).

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A348331 Lexicographically earliest sequence of positive integers such that for any n > 0, the sum of the indices k < n such that a(k) = a(n) is less than or equal to n.

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A114958 a(n) = 6*2^(n+1) - 5*(n+1) - 4.

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A266926 a(0)=0, a(1)=1, a(2)=10; for n>2, a(n) = concat(a(1), ..., a(n-1)).

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A176414 Expansion of (7+8x)/(1+2x).

A114958 a(n) = 62^(n+1) - 5(n+1) - 4.