A005056 -id:A005056 - cp's OEIS frontend

A083318 a(0) = 1; for n>0, a(n) = 2^n + 1.

1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649

Offset: 0

Paul Barry, Apr 25 2003

Inverse binomial transform of A005056.
Also, A000533 interpreted as binary numbers, written in base 10. Numbers whose representation in base 2 is has n+1 digits and the digit "1" is the initial and final digit and if n>1 then the internal digits are "0" (see example). - Omar E. Pol, Feb 24 2008
a(n) equals the number of ternary sequences of length n such that no two consecutive terms differ by 1. - David Nacin, May 31 2017

			From _Omar E. Pol_, Feb 24 2008: (Start)
------------------------------
n .... a(n) .. a(n) in base 2
------------------------------
0 ..... 1 ..... 1
1 ..... 3 ..... 11
2 ..... 5 ..... 101
3 ..... 9 ..... 1001
4 .... 17 ..... 10001
5 .... 33 ..... 100001
6 .... 65 ..... 1000001
7 ... 129 ..... 10000001
8 ... 257 ..... 100000001
9 ... 513 ..... 1000000001
(End)
G.f. = 1 + 3*x + 5*x^2 + 9*x^3 + 17*x^4 + 33*x^5 + 65*x^6 + 129*x^7 + ... - _Michael Somos_, Jun 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ramon Carbó-Dorca, Boolean Hypercubes: The Origin of a Tagged Recursive Logic and the Limits of Artificial Intelligence, Universitat de Girona (Spain, 2020).
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Except for the leading term, the same as A000051.

Cf. A000533, A083319, A132749.

Concatenation([1], List([1..40], n-> 2^n +1)); # G. C. Greubel, Nov 20 2019

[2^n+1-0^n : n in [0..40]]; // Vincenzo Librandi, Sep 01 2011

seq(`if`(n=0, 1, 2^n + 1), n=0..40); # G. C. Greubel, Nov 20 2019

Join[{1},2^Range[40]+1] (* Harvey P. Dale, May 17 2013 *)

{a(n) = if( n<1, n==0, 2^n + 1)}; /* Michael Somos, Jun 04 2016 */

[1]+[2^n +1 for n in (1..40)] # G. C. Greubel, Nov 20 2019

a(n) = 2^n + 1^n - 0^n.
G.f.: (1-2*x^2)/((1-x)*(1-2x)).
E.g.f.: exp(2*x) + exp(x) - exp(0).
a(n) = Sum_{k=0..n} 0^(k*(n-k))*2^(n-k). - Paul Barry, Feb 09 2005
a(n) = Min{m: A008687(m) = n+1}. - Reinhard Zumkeller, Jul 25 2006
Row sums of triangle A132749; = binomial transform of [1, 2, 0, 2, 0, 2, 0, 2, ...]. - Gary W. Adamson, Aug 28 2007
A020650(a(n)) = 1. - Yosu Yurramendi, Jun 01 2016

Edited by N. J. A. Sloane, Sep 28 2007

A155588 a(n) = 5^n + 2^n - 1^n.

Original entry on oeis.org

1, 6, 28, 132, 640, 3156, 15688, 78252, 390880, 1953636, 9766648, 48830172, 244144720, 1220711316, 6103532008, 30517610892, 152587956160, 762939584196, 3814697527768, 19073486852412, 95367432689200, 476837160300276

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1430
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A074501, A074600, A005057, A005056, A099393.

CoefficientList[Series[1/(1 - 5 x) + 1/(1 - 2 x) - 1/(1 - x), {x, 0, 21}], x] (* Michael De Vlieger, Mar 02 2022 *)

a(n)=5^n+2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-5*x)+1/(1-2*x)-1/(1-x).
E.g.f.: e^(5*x)+e^(2*x)-e^x.
a(n) = 7*a(n-1)-10*a(n-2)-4 with a(0)=1, a(1)=6. - Vincenzo Librandi, Jul 21 2010
a(n) = A074600(n)-1. - R. J. Mathar, Mar 10 2022

A155590 a(n) = 7^n + 2^n - 1.

Original entry on oeis.org

1, 8, 52, 350, 2416, 16838, 117712, 823670, 5765056, 40354118, 282476272, 1977328790, 13841291296, 96889018598, 678223089232, 4747561542710, 33232930635136, 232630514118278, 1628413598172592, 11398895185897430, 79792266298660576, 558545864085381158, 3909821048587182352

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23,14).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590.

Table[7^n + 2^n - 1, {n, 0, 25}] (* Paolo Xausa, Jul 30 2024 *)

a(n)=7^n+2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-7*x)+1/(1-2*x)-1/(1-x).
E.g.f.: exp(7*x)+exp(2*x)-exp(x).
a(n) = 9*a(n-1)-14*a(n-2)-6 with a(0) = 1, a(1) = 8. - Vincenzo Librandi, Jul 21 2010
a(n) = A074602(n)-1. - R. J. Mathar, Mar 10 2022

A155592 8^n+2^n-1^n.

Original entry on oeis.org

1, 9, 67, 519, 4111, 32799, 262207, 2097279, 16777471, 134218239, 1073742847, 8589936639, 68719480831, 549755822079, 4398046527487, 35184372121599, 281474976776191, 2251799813816319, 18014398509744127, 144115188076380159

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590.

Table[8^n+2^n-1^n,{n,0,30}] (* or *) LinearRecurrence[{11,-26,16},{1,9,67},30] (* Harvey P. Dale, Jul 09 2017 *)

a(n)=8^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-8*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(8*x)+e^(2*x)-e^x.
a(n)=10*a(n-1)-16*a(n-2)-7 with a(0)=1, a(1)=9 - Vincenzo Librandi, Jul 21 2010
a(n) = A074603(n)-1. - R. J. Mathar, Mar 10 2022

A155593 a(n) = 9^n + 2^n - 1.

Original entry on oeis.org

1, 10, 84, 736, 6576, 59080, 531504, 4783096, 43046976, 387421000, 3486785424, 31381061656, 282429540576, 2541865836520, 22876792471344, 205891132127416, 1853020188917376, 16677181699797640, 150094635297261264

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-29,18).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592.

Table[9^n+2^n-1,{n,0,30}] (* or *) LinearRecurrence[{12,-29,18},{1,10,84},30] (* Harvey P. Dale, Sep 09 2022 *)

a(n)=9^n+2^n-1^n \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-9*x)+1/(1-2*x)-1/(1-x).
E.g.f.: e^(9*x)+e^(2*x)-e^x.
a(n) = 11*a(n-1)-18*a(n-2)-8 with a(0)=1, a(1)=10. - Vincenzo Librandi, Jul 21 2010
a(n) = A074604(n)-1. - R. J. Mathar, Mar 10 2022

A155594 10^n+2^n-1.

Original entry on oeis.org

1, 11, 103, 1007, 10015, 100031, 1000063, 10000127, 100000255, 1000000511, 10000001023, 100000002047, 1000000004095, 10000000008191, 100000000016383, 1000000000032767, 10000000000065535, 100000000000131071

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592, A155593.

LinearRecurrence[{13,-32,20},{1,11,103},20] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=10^n+2^n-1^n \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-10*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(10*x)+e^(2*x)-e^x.
a(n)=12*a(n-1)-20*a(n-2)-9 with a(0)=1, a(1)=11 - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=11, a(2)=103, a(n)=13*a(n-1)-32*a(n-2)+20*a(n-3). - Harvey P. Dale, Mar 07 2015
a(n) = A050621(n+1)-1. - R. J. Mathar, Mar 10 2022

A155589 a(n) = 6^n + 2^n - 1.

Original entry on oeis.org

1, 7, 39, 223, 1311, 7807, 46719, 280063, 1679871, 10078207, 60467199, 362799103, 2176786431, 13060702207, 78364180479, 470185017343, 2821109972991, 16926659575807, 101559956930559, 609359740534783, 3656158441111551, 21936950642475007, 131621703846461439

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A074501, A074600, A005057, A005056, A099393, A155588.

Table[6^n + 2^n - 1, {n, 0, 25}] (* Paolo Xausa, Jul 30 2024 *)

a(n)=6^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-6*x)+1/(1-2*x)-1/(1-x).
E.g.f.: exp(6*x)+exp(2*x)-exp(x).
a(n) = 8*a(n-1)-12*a(n-2)-5 with a(0) = 1, a(1) = 7. - Vincenzo Librandi, Jul 21 2010

A155595 11^n+2^n-1.

Original entry on oeis.org

1, 12, 124, 1338, 14656, 161082, 1771624, 19487298, 214359136, 2357948202, 25937425624, 285311672658, 3138428380816, 34522712152122, 379749833599624, 4177248169448418, 45949729863637696, 505447028499424842

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (14,-35,22).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592, A155593, A155594.

a(n)=11^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-11*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(11*x)+e^(2*x)-e^x.

a(n)=13*a(n-1)-22*a(n-2)-10 with a(0)=1, a(1)=12 - Vincenzo Librandi, Jul 21 2010

A265417 Rectangular array T(n,m), read by upward antidiagonals: T(n,m) is the number of difunctional (regular) binary relations between an n-element set and an m-element set.

Original entry on oeis.org

2, 4, 4, 8, 12, 8, 16, 34, 34, 16, 32, 96, 128, 96, 32, 64, 274, 466, 466, 274, 64, 128, 792, 1688, 2100, 1688, 792, 128, 256, 2314, 6154, 9226, 9226, 6154, 2314, 256, 512, 6816, 22688, 40356, 48032, 40356, 22688, 6816, 512, 1024, 20194, 84706, 177466, 245554, 245554, 177466, 84706, 20194, 1024, 2048, 60072, 320168

Offset: 1

Jasha Gurevich, Dec 08 2015

T(n,m) is the number of difunctional (regular) binary relations between an n-element set and an m-element set.
From Petros Hadjicostas, Feb 09 2021: (Start)
From Knopfmacher and Mays (2001): "Let G be a labeled graph, with edge set E(G) and vertex set V(G). A composition of G is a partition of V(G) into vertex sets of connected induced subgraphs of G." "We will denote by C(G) the number of distinct compositions that exist for a given graph G."
By Theorem 10 in Knofmacher and Mays (2001), T(n,m) = C(K_{n,m}) = Sum_{i=1..n+1} A341287(n,i)*i^m, where K_{n,m} is the complete bipartite graph with n+m vertices and n*m edges. For values of T(n,m), see the table on p. 10 of the paper.
Huq (2007) reproved the result using different methodology and derived the bivariate e.g.f. of T(n,m). (End)

			Array T(n,m) (with rows n >= 1 and columns m >= 1) begins:
    2      4      8      16       32        64        128         256 ...
    4     12     34      96      274       792       2314        6816 ...
    8     34    128     466     1688      6154      22688       84706 ...
   16     96    466    2100     9226     40356     177466      788100 ...
   32    274   1688    9226    48032    245554    1251128     6402586 ...
   64    792   6154   40356   245554   1444212    8380114    48510036 ...
  128   2314  22688  177466  1251128   8380114   54763088   354298186 ...
  256   6816  84706  788100  6402586  48510036  354298186  2540607060 ...
  512  20194 320168 3541066 33044432 281910994 2288754728 18082589146 ...
  ...

Jasha Gurevich, Table of n, a(n) for n = 1..300
Chris Brink, Wolfram Kahl, and Gunther Schmidt, Relational Methods in Computer Science, Springer Science & Business Media, 1997, p. 200.
Jasha Gurevich, Full list of formulas for correspondences (binary relations).
Amiqul Huq, Compositions of graphs revisited, Vol. 14 (2007), Article N15.
A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumerations, Integers, Vol. 1 (2001), Article A4. (See p. 10 for the table and p. 8 for a formula.)
J. Riguet, Relations binaires, fermetures, correspondances de Galois, Bulletin de la Société Mathématique de France, Vol. 76 (1948), 114-155.
Wikipedia, Binary relation.

Cf. A005056 (1st line or column ?), A014235 (diagonal ?), A341287.

sum((Stirling2(m, i-1)*factorial(i)+Stirling2(m, i)*factorial(i+1)+Stirling2(m, i+1)*factorial(i+1))*Stirling2(n, i), i = 1 .. n)

T(n, m) = sum(i=1,n, (stirling(m, i-1,2)*i! + stirling(m, i,2)*(i+1)! + stirling(m, i+1,2)*(i+1)!)*stirling(n, i,2)); \\ Michel Marcus, Dec 10 2015

T(n, m) = Sum_{i=1..n} (Stirling2(m, i-1)*i! + Stirling2(m, i)*(i+1)! + Stirling2(m, i+1)*(i+1)!)*Stirling2(n, i).
From Petros Hadjicostas, Feb 09 2021: (Start)
T(n,m) = Sum_{i=1..n+1} A341287(n,i)*i^m = Sum_{i=1..m+1} A341287(m,i)*i^n. (See Knopfmacher and Mays (2001) and Huq (2007).)
Bivariate e.g.f.: Sum_{n,m >= 1} T(n,m)*(x^n/n!)*(y^m/m!) = exp((exp(x) - 1)*(exp(y) - 1) + x + y) - exp(x) - exp(y) + 1. (This is a modification of Eq. (7) in Huq (2007), p. 4.) (End)

A346990 Numbers occurring as divisors of 2^k + 3^k, k > 0.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 25, 31, 35, 37, 41, 49, 55, 59, 61, 73, 77, 79, 83, 89, 97, 103, 107, 109, 113, 121, 125, 127, 131, 137, 149, 151, 155, 157, 169, 173, 175, 179, 181, 193, 199, 217, 223, 227, 229, 233, 241, 245, 251, 257, 271, 275, 277, 281, 289, 295, 313

Offset: 1

Hugo Pfoertner, Aug 11 2021

nonn

If n is a term, then so are all divisors of n. - Robert Israel, Dec 08 2022

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

Cf. A007689, A005056.

filter:= proc(n) local v;
   if igcd(n,6) <> 1 then return false fi;
   q:= 3/2 mod n;
   traperror(NumberTheory:-ModularLog(-1,q,n)) <> lasterror
end proc:
filter(1):= true:
select(filter, [$1..400]); # Robert Israel, Dec 08 2022

cp's OEIS Frontend

A083318 a(0) = 1; for n>0, a(n) = 2^n + 1.

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A155588 a(n) = 5^n + 2^n - 1^n.

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A155590 a(n) = 7^n + 2^n - 1.

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A155592 8^n+2^n-1^n.

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A155593 a(n) = 9^n + 2^n - 1.

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A155594 10^n+2^n-1.

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A155589 a(n) = 6^n + 2^n - 1.

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A155595 11^n+2^n-1.

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