A132749 -id:A132749 - cp's OEIS frontend

A132750 A132749 * [1, 2, 3, ...] = A007318 * A065190.

1, 4, 9, 21, 49, 113, 257, 577, 1281, 2817, 6145, 13313, 28673, 61441, 131073, 278529, 589825, 1245185, 2621441, 5505025, 11534337, 24117249, 50331649, 104857601, 218103809, 452984833, 939524097, 1946157057, 4026531841

Offset: 0

Gary W. Adamson, Aug 28 2007

Equals double binomial transform of [1, 2, -3, 7, -15, 31, -63, 127, -255, ...]. - Gary W. Adamson, Jul 23 2008

For n >= 1, also the number of cliques in the n-hypercube graph Q_n. - Eric W. Weisstein, Mar 31 2017

			a(3) = 21 = (1, 3, 3, 1) dot (1, 3, 2, 5) = (1 + 9 + 6 + 5) = 21; where A065190 = (1, 3, 2, 5, 4, 7, 6, 9, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Jelinek, T. Mansour, M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Lemma 4.21 (sequence starting 1, 1, 2, 4, 9, 21, .... with offset 0).
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Hypercube Graph
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A065190, A132749, A135224.

Concatenation([1], List([1..30], n-> n*2^(n-1) + 2^n + 1 )); # G. C. Greubel, Nov 20 2019

[n*2^(n-1) + 2^n + 1 - 0^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 26 2014

A132750:=n->n*2^(n-1)+2^n+1-0^n: seq(A132750(n), n=0..30); # Wesley Ivan Hurt, Sep 26 2014

Join[{1}, Table[n*2^(n-1) +2^n +1, {n, 30}]] (* Wesley Ivan Hurt, Sep 26 2014 *)
Join[{1}, LinearRecurrence[{5,-8,4}, {4,9,21}, 30]] (* Vincenzo Librandi, Apr 01 2017 *)

vector(31, n, if(n==1, 1, (n-1)*2^(n-2) + 2^(n-1) + 1)) \\ G. C. Greubel, Nov 20 2019

[1]+[n*2^(n-1) + 2^n + 1 for n in (1..30)] # G. C. Greubel, Nov 20 2019

A132749 as an infinite lower triangular matrix * vector [1, 2, 3, ...]. Binomial transform of A065190 (with an incorrect offset)
Row sums of triangle A135224. - Gary W. Adamson, Nov 23 2007
G.f.: (1-x-3*x^2+4*x^3)/((1-x)*(1-2*x)^2). - Colin Barker, Aug 09 2012
a(n) = n*2^(n-1) + 2^n + 1 - 0^n. - Tim Smith, Sep 25 2014
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Wesley Ivan Hurt, Sep 26 2014
E.g.f.: -1 + exp(x) + (1+x)*exp(2*x). - G. C. Greubel, Nov 20 2019

A132752 Triangle T(n, k) = 2*A132749(n, k) - 1, read by rows.

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 3, 5, 5, 1, 3, 7, 11, 7, 1, 3, 9, 19, 19, 9, 1, 3, 11, 29, 39, 29, 11, 1, 3, 13, 41, 69, 69, 41, 13, 1, 3, 15, 55, 111, 139, 111, 55, 15, 1, 3, 17, 71, 167, 251, 251, 167, 71, 17, 1

Offset: 0

Gary W. Adamson, Aug 28 2007

			First few rows of the triangle are:
  1;
  3,  1;
  3,  3,  1;
  3,  5,  5,  1;
  3,  7, 11,  7,  1;
  3,  9, 19, 19,  9,  1;
  3, 11, 29, 39, 29, 11, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A109128, A132749, A132753.

A132752:= func< n,k | k eq n select 1 else k eq 0 select 3 else 2*Binomial(n,k) -1 >;
[A132752(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_]:= If[k==n, 1, If[k==0, 3, 2*Binomial[n, k] -1 ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A132752(n,k): return 1 if k==n else 3 if k==0 else 2*binomial(n,k) -1
flatten([[A132752(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

T(n, k) = 2*A132749(n, k) - 1, an infinite lower triangular matrix.
From G. C. Greubel, Feb 16 2021: (Start)
T(n, k) = A109128(n, k) with T(n, 0) = 3.
Sum_{k=0..n} T(n, k) = 2^(n+1) -n +1 -2*[n=0] = A132753(n) - 2*[n=0]. (End)

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

Original entry on oeis.org

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

cp's OEIS Frontend

A132750 A132749 * [1, 2, 3, ...] = A007318 * A065190.

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A132752 Triangle T(n, k) = 2*A132749(n, k) - 1, read by rows.

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

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GAP

Magma

Maple

Mathematica

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