A053208 -id:A053208 - cp's OEIS frontend

A053209 Row sums of A051598.

1, 5, 14, 32, 68, 140, 284, 572, 1148, 2300, 4604, 9212, 18428, 36860, 73724, 147452, 294908, 589820, 1179644, 2359292, 4718588, 9437180, 18874364, 37748732, 75497468, 150994940, 301989884, 603979772, 1207959548, 2415919100

Offset: 0

Asher Auel, Dec 14 1999

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A051598, A053208.
Cf. A083329, A131051.
Cf. A048491.

I:=[5,14]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 03 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)^2)/((1-x)*(1-2*x))); // Marius A. Burtea, Oct 15 2019

Join[{1}, LinearRecurrence[{3, -2}, {5, 14}, 50]] (* G. C. Greubel, Sep 03 2018 *)

m=30; v=concat([5,14], vector(m-2)); for(n=3, m, v[n] = 3*v[n-1] -2*v[n-2]); concat([1], v) \\ G. C. Greubel, Sep 03 2018

a(0) = 1, a(1) = 5, a(n+1) = 2*a(n) + 4, for n >= 1.
a(n) = 9*2^(n-1) - 4, n >= 1.
a(n) =  4*n + Sum[i = 0, n - 1] a(i). - Jon Perry, Nov 20 2012
a(n) = A048491(n)/2, n>0. - Philippe Deléham, Apr 15 2013
G.f.: (1+x)^2/((1-x)*(1-2*x)). - Philippe Deléham, Apr 15 2013
a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=5, a(2)=14. - Philippe Deléham, Apr 15 2013
E.g.f.: (1 - 8*exp(x) + 9*exp(2*x))/2. - Stefano Spezia, Sep 28 2022

A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 7, 10, 7, 15, 24, 24, 15, 31, 52, 59, 52, 31, 63, 108, 129, 129, 108, 63, 127, 220, 269, 284, 269, 220, 127, 255, 444, 549, 594, 594, 549, 444, 255, 511, 892, 1109, 1214, 1245, 1214, 1109, 892, 511, 1023, 1788, 2229, 2454, 2547, 2547, 2454

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,3,7,15,31,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202873 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202767 for characteristic polynomials of principal submatrices of M.

			Northwest corner:
1.....3.....7...15...31.....63
3....10....24...52...108...220
7....24....59..129...269...549
15...52...129..284...594..1214
31...108..269..594..1245..2547

Cf. A202767.

s[k_] := -1 + 2^k;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A000295, Eulerian *)
Table[m[1, j], {j, 1, 12}]    (* A000225 *)
Table[m[2, j], {j, 1, 12}]    (* A053208 *)

A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 10, 10, 7, 15, 22, 24, 22, 15, 31, 46, 52, 52, 46, 31, 63, 94, 108, 112, 108, 94, 63, 127, 190, 220, 232, 232, 220, 190, 127, 255, 382, 444, 472, 480, 472, 444, 382, 255, 511, 766, 892, 952, 976, 976, 952, 892, 766, 511, 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

T(n, k) is the expanded 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 with 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.

			Triangle begins:
  0;
  1,     1;
  3,     4,    3;
  7,    10,   10,    7;
  15,   22,   24,   22,   15;
  31,   46,   52,   52,   46,   31;
  63,   94,  108,  112,  108,   94,   63;
 127,  190,  220,  232,  232,  220,  190,  127;
 255,  382,  444,  472,  480,  472,  444,  382,  255;
 511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
  ...

Column k=0 gives A000225(n-1).
Column k=1 gives A033484(n-2).
Column k=2 gives A053208(n-3).
Cf. A005061, A359200.

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

T[n_, k_] := 2^n - 2^(n - k - 1) - 2^k; Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

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

Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)

A053207 Rows of triangle formed using Pascal's rule except begin n-th row with n and end it with n+1.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 7, 6, 7, 11, 13, 13, 8, 9, 18, 24, 26, 21, 10, 11, 27, 42, 50, 47, 31, 12, 13, 38, 69, 92, 97, 78, 43, 14, 15, 51, 107, 161, 189, 175, 121, 57, 16, 17, 66, 158, 268, 350, 364, 296, 178, 73, 18, 19, 83, 224, 426, 618, 714, 660, 474, 251, 91, 20

Offset: 0

Asher Auel, Dec 13 1999

			Triangle begins:
  0;
  1,  2;
  3,  3,  4;
  5,  6,  7,  6;
  7, 11, 13, 13, 8;
  ...

Row sums give A053208.

Cf. A007318 (Pascal's triangle)

cp's OEIS Frontend

A053209 Row sums of A051598.

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A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals.

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A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

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A053207 Rows of triangle formed using Pascal's rule except begin n-th row with n and end it with n+1.

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