A145654 -id:A145654 - cp's OEIS frontend

A121173 Sequence S with property that for n in S, a(n) = a(1) + a(2) +...+ a(n-1) and for n not in S, a(n) = n+1.

2, 2, 4, 8, 6, 22, 8, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18, 2026, 20, 4072, 22, 8166, 24, 16356, 26, 32738, 28, 65504, 30, 131038, 32, 262108, 34, 524250, 36, 1048536, 38, 2097110, 40, 4194260, 42, 8388562, 44, 16777168, 46, 33554382

Offset: 1

Max Alekseyev, Aug 15 2006

nonn

a(1)=1 cannot happen, so the sequence S starts with a(1)=2.
Note that a(n)=a(1)+a(2)+...+a(n-1) can hold even if n is not in S. The smallest example is n=3.
All terms are even. - Reinhard Zumkeller, Nov 06 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A105753, A121053, A121174, A121175, A145654.

a121173 n = a121173_list !! (n-1)
a121173_list = f 1 [] where
   f x ys = y : f (x + 1) (y : ys) where
     y = if x `elem` ys then sum ys else x + 1
-- Reinhard Zumkeller, Nov 06 2013

s={2};Do[If[MemberQ[s,n],m=Total[s],m=n+1];AppendTo[s,m],{n,2,46}];s (* James C. McMahon, Oct 13 2024 *)

from itertools import count, islice
def agen(): # generator of terms
    S, s, an = {2}, 2, 2
    for n in count(2):
        yield an
        an = s if n in S else n+1
        s += an
        S.add(an)
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2024

a(2*n) = A145654(n+1). - Reinhard Zumkeller, Nov 06 2013
a(2*n+1) = 2*n+2.
From Colin Barker, Jan 30 2016: (Start)
a(n) = 2*(2^(n/2+1)-2)-n for n even.
a(n) = n+1 for n odd.
a(n) = -a(n-1)+3*a(n-2)+3*a(n-3)-2*a(n-4)-2*a(n-5) for n>5.
G.f.: 2*x*(1+2*x) / ((1-x)*(1+x)^2*(1-2*x^2)). (End)
E.g.f.: (x - 4)*cosh(x) + 4*cosh(sqrt(2)*x) + (1 - x)*sinh(x). - Stefano Spezia, Oct 14 2024

A350770 Triangle read by rows: T(n, k) = 2^(n-k-1) + 2^k - 2, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 7, 4, 4, 7, 15, 8, 6, 8, 15, 31, 16, 10, 10, 16, 31, 63, 32, 18, 14, 18, 32, 63, 127, 64, 34, 22, 22, 34, 64, 127, 255, 128, 66, 38, 30, 38, 66, 128, 255, 511, 256, 130, 70, 46, 46, 70, 130, 256, 511, 1023, 512, 258, 134, 78, 62, 78, 134, 258, 512, 1023, 2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047

Offset: 1

Ambrosio Valencia-Romero, Jan 14 2022

T(n, k) is the number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies without the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.

The sum of the products of T(n, k) and binomial(n-1,k) for 0 <= k <= n-1 equals 2*A001047(n-1). For instance, for n = 3, T(3, k) returns 3, 2, and 3 and binomial(3-1,k) returns 1, 2, and 1 for k = 0, 1, and 2, respectively. Then 3*1 + 2*2 + 3*1 = 2*A001047(3-1) = 2*5 = 10. Similarly, for n = 4, the result yields 7*1 + 4*3 + 4*3 + 7*1 = 2*A001047(4-1) = 2*19 = 38.

			Triangle begins:
     0;
     1,    1;
     3,    2,   3;
     7,    4,   4,   7;
    15,    8,   6,   8,  15;
    31,   16,  10,  10,  16, 31;
    63,   32,  18,  14,  18, 32, 63;
   127,   64,  34,  22,  22, 34, 64, 127;
   255,  128,  66,  38,  30, 38, 66, 128, 255;
   511,  256, 130,  70,  46, 46, 70, 130, 256, 511;
  1023,  512, 258, 134,  78, 62, 78, 134, 258, 512, 1023;
  2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50).
Ambrosio Valencia-Romero, Strategy Dynamics in Collective Systems Design, Ph.D. Thesis, Stevens Institute of Technology (Hoboken, 2021). [Table 5.4, page 67]
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).

Column k=0 gives A000225(n-1).
Row sums give A145654.
Cf. A001047.

T := n -> seq(2^(n - k - 1) + 2^k - 2, k = 0 .. n - 1);
seq(T(n), n=1..12);

T(n, k) = 2^(n-k-1) + 2^k - 2 \\ Andrew Howroyd, May 06 2023

T(n, k) = 2^(n-k-1) + 2^k - 2.

A271698 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 8, 1, 0, 0, 2, 28, 22, 1, 0, 0, 2, 72, 182, 52, 1, 0, 0, 2, 164, 974, 864, 114, 1, 0, 0, 2, 352, 4174, 8444, 3474, 240, 1, 0, 0, 2, 732, 15782, 61464, 57194, 12660, 494, 1, 0, 0, 2, 1496, 55286, 373940, 660842, 332528, 43358, 1004, 1, 0

Offset: 0

Peter Luschny, Apr 12 2016

			Triangle starts:
1,
1, 0,
0, 1, 0,
0, 2, 1, 0,
0, 2, 8, 1, 0,
0, 2, 28, 22, 1, 0,
0, 2, 72, 182, 52, 1, 0,
0, 2, 164, 974, 864, 114, 1, 0

Peter Luschny, Extensions of the binomial

A000255 (row sums), compare A028296 for alternating rows sums, A145654 and A005803 (diag. n,n-2).

Cf. A173018.

A271698 := (n,k) -> add(binomial(-j,-n)*combinat:-eulerian1(j,k), j=0..n):
seq(seq(A271698(n, k), k=0..n), n=0..10);

Mathematica
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A121173 Sequence S with property that for n in S, a(n) = a(1) + a(2) +...+ a(n-1) and for n not in S, a(n) = n+1.

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A350770 Triangle read by rows: T(n, k) = 2^(n-k-1) + 2^k - 2, 0 <= k <= n-1.

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A271698 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

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Mathematica