A132925 -id:A132925 - cp's OEIS frontend

A006127 a(n) = 2^n + n.

1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, 16398, 32783, 65552, 131089, 262162, 524307, 1048596, 2097173, 4194326, 8388631, 16777240, 33554457, 67108890, 134217755, 268435484, 536870941, 1073741854, 2147483679, 4294967328, 8589934625

Offset: 0

N. J. A. Sloane

For numbers m=n+2^n such that equation x=2^(m-x) has solution x=2^n, see A103354. - Zak Seidov, Mar 23 2005
Primes of the form x^x+1 must be of the form 2^2^(a(n))+1, that is, Fermat number F_(a(n)) (Sierpiński 1958). - David W. Wilson, May 08 2005
a(n) = n-th Mersenne number + n + 1 = A000225(n) + n + 1. Partial sums of a(n) are A132925(n+1). - Jaroslav Krizek, Oct 16 2009
Intersection of A188916 and A188917: A188915(a(n)) = (2^n)^2 = 2^(2*n) = A000302(n). - Reinhard Zumkeller, Apr 14 2011
a(n) is also the number of all connected subtrees of a star tree, having n leaves. The star tree is a tree, where all n leaves are connected to one parent P. - Viktar Karatchenia, Feb 29 2016

			From _Viktar Karatchenia_, Feb 29 2016: (Start)
a(0) = 1. There are n=0 leaves, it is a trivial tree consisting of a single parent node P.
a(1) = 3. There is n=1 leaf, the tree is P-A, the subtrees are: 2 singles: P, A; 1 pair: P-A; 2+1 = 3 subtrees in total.
a(2) = 6. When n=2, the tree is P-A P-B, the subtrees are: 3 singles: P, A, B; 2 pairs: P-A, P-B; 1 triple: A-P-B (the whole tree); 3+2+1 = 6.
a(3) = 11. For n=3 leaf nodes, the tree is P-A P-B P-C, the subtrees are: 4 singles: P, A, B, C; 3 pairs: P-A, P-B, P-C; 3 triples: A-P-B, A-P-C, B-P-C; 1 quad: P-A P-B P-C (the whole tree); 4+3+3+1 = 11 in total.
a(4) = 20. For n=4 leaves, the tree is P-A P-B P-C P-D, the subtrees are: 5 singles: P, A, B, C, D; 4 pairs: P-A, P-B, P-C, P-D; 6 triples: A-P-B, A-P-C, B-P-C, A-P-D, B-P-D, C-P-D; 4 quads: P-A P-B P-C, P-A P-B P-D, P-A P-C P-D, P-B P-C P-D; the whole tree counts as 1; 5+4+6+4+1 = 20.
In general, for n leaves, connected to the parent node P, there will be: (n+1) singles; (n, 1) pairs; (n, 2) triples; (n, 3) quads; ... ; (n, n-1) subtrees having (n-1) edges; 1 whole tree, having all n edges. Thus, the total number of all distinct subtrees is: (n+1) + (n, 1) + (n, 2) + (n, 3) + ... + (n, n-1) + 1 = (n + (n, 0)) + (n, 1) + (n, 2) + (n, 3) + ... + (n, n-1) + (n, n) = n + (sum of all binomial coefficients of size n) = n + (2^n). (End)

John H. Conway, R. K. Guy, The Book of Numbers, Copernicus Press, p. 84.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 435
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind
Eric Weisstein's World of Mathematics, Star Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A135227, A000079, A052944; A000051 (first differences).

Cf. A000325.

a006127 n = a000079 n + n
a006127_list = s [1] where
   s xs = last xs : (s $ zipWith (+) [1..] (xs ++ reverse xs))
Reinhard Zumkeller, May 19 2015, Feb 05 2011

A006127:=(-1+z+z**2)/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Table[2^n + n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
Table[BitXOr(i, 2^i), {i, 1, 100}] (* Peter Luschny, Jun 01 2011 *)
LinearRecurrence[{4,-5,2},{1,3,6},40] (* Harvey P. Dale, Feb 08 2023 *)

a(n)=1<Charles R Greathouse IV, Jul 19 2011

print([2**n + n for n in range(34)]) # Karl V. Keller, Jr., Aug 18 2020

def A006127(n): return (1<Chai Wah Wu, Jan 11 2023

Row sums of triangle A135227. - Gary W. Adamson, Nov 23 2007
Partial sums of A094373. G.f.: (1-x-x^2)/((1-x)^2(1-2x)). - Paul Barry, Aug 05 2004
Binomial transform of [1,2,1,1,1,1,1,...]. - Franklin T. Adams-Watters, Nov 29 2006
a(n) = 2*a(n-1) - n + 2 (with a(0)=1). - Vincenzo Librandi, Dec 30 2010
E.g.f.: exp(x)*(exp(x) + x). - Stefano Spezia, Dec 10 2021

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

A132924 Triangle read by columns, 2^(n-1) followed by (2^(n-1) + 1), (2^(n-1) + 2), (2^(n-1) + 3), ...

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 8, 5, 5, 6, 9, 16, 6, 6, 7, 10, 17, 32, 7, 7, 8, 11, 18, 33, 64, 8, 8, 9, 12, 19, 34, 65, 128, 9, 9, 10, 13, 20, 35, 66, 129, 256, 10, 10, 11, 14, 21, 36, 67, 130, 257, 512, 11, 11, 12, 15, 22, 37, 68, 131, 258, 513, 1024, 12, 12, 13, 16, 23, 38, 69, 132

Offset: 1

Gary W. Adamson, Sep 05 2007

Row sums = A132925. Right border = 2^0, 2^1, 2^2, ...

			First few rows of the triangle:
  1;
  2, 2;
  3, 3, 4;
  4, 4, 5, 8;
  5, 5, 6, 9, 16;
  6, 6, 7, 10, 17, 32;
  7, 7, 8, 11, 18, 33, 64;
  ...

Cf. A132925.

A132924 := proc(n,m) n+2^m-m ; end: seq(seq(A132924(n,m),m=0..n),n=0..13) ; # R. J. Mathar, Oct 23 2009

More terms from R. J. Mathar, Oct 23 2009

A235804 Rectangular array read by upward antidiagonals: A(n,k) = n-2+k*2^(n-3), n>=3, k>=0.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 7, 6, 4, 5, 12, 11, 8, 5, 6, 21, 20, 15, 10, 6, 7, 38, 37, 28, 19, 12, 7, 8, 71, 70, 53, 36, 23, 14, 8, 9, 136, 135, 102, 69, 44, 27, 16, 9, 10, 265, 264, 199, 134, 85, 52, 31, 18, 10, 11, 522, 521, 392, 263, 166, 101, 60, 35, 20, 11

Offset: 3

L. Edson Jeffery, Jan 16 2014

Row index n begins with 3, column index k begins with 0.

Conjectured equivalence classes associated with the row entries of A233332.

			Array begins:
 1,   2,    3,    4,    5,    6,    7,    8,    9,   10, ...
 2,   4,    6,    8,   10,   12,   14,   16,   18,   20, ...
 3,   7,   11,   15,   19,   23,   27,   31,   35,   39, ...
 4,  12,   20,   28,   36,   44,   52,   60,   68,   76, ...
 5,  21,   37,   53,   69,   85,  101,  117,  133,  149, ...
 6,  38,   70,  102,  134,  166,  198,  230,  262,  294, ...
 7,  71,  135,  199,  263,  327,  391,  455,  519,  583, ...
 8, 136,  264,  392,  520,  648,  776,  904, 1032, 1160, ...
 9, 265,  521,  777, 1033, 1289, 1545, 1801, 2057, 2313, ...
10, 522, 1034, 1546, 2058, 2570, 3082, 3594, 4106, 4618, ...

Cf. A000295, A132925 (conjectured antidiagonal sums), A233332.

Conjecture: G.f. for row n is F_n(x) = ((n-2)+(2^(n-3)-(n-2))*x)/(1-x)^2 = ((n-2)+(2^(n-3)-(n-3)-1)*x)/(1-x)^2 = ((n-2)+A000295(n-3)*x)/(1-x)^2, n>=3.

Conjecture: G.f. for column k is G_k(x) = (k+1-2*(k+1)*x+k*x^2)/((1-2*x)*(1-x)^2), k>=0.

cp's OEIS Frontend

A006127 a(n) = 2^n + n.

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A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

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A132924 Triangle read by columns, 2^(n-1) followed by (2^(n-1) + 1), (2^(n-1) + 2), (2^(n-1) + 3), ...

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A235804 Rectangular array read by upward antidiagonals: A(n,k) = n-2+k*2^(n-3), n>=3, k>=0.

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