A168316 -id:A168316 - cp's OEIS frontend

A168317 Eigensequence of triangle A168316.

1, 1, 3, 6, 16, 39, 103, 263, 690, 1791, 4693, 12247, 32073, 83869, 219598, 574658, 1504540, 3938272, 10310703

Offset: 1

Gary W. Adamson, Nov 22 2009

Conjectured convergent of a(n)/a(n-1) = phi^2 = 2.6180339... E.g.: a(19)/a(18) = 10310703/3938272 = 2.6180779....

Equals lim_{n->oo} M^2, the left shifted vector considered as a sequence where M = triangle A168316 as an infinite lower triangular matrix, then shifted down one row, and inserting a "1" at top.

A168318 Triangle read by rows, A168316 * its diagonalized eigensequence, A168317.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 2, 3, 0, 1, 3, 6, 6, 0, 0, 2, 9, 12, 16, 0, 0, 2, 12, 18, 32, 39, 0, 0, 1, 9, 24, 48, 78, 103, 0, 0, 1, 9, 30, 64, 117, 206, 263, 0, 0, 0, 6, 24, 80, 156, 309, 526, 690, 0, 0, 0, 6, 24, 96, 195, 412, 789, 1380, 1791

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = A168317: (1, 1, 3, 6, 16, 39, 103, 263,...).
Rightmost diagonal = A168317 prefaced with a 1.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 1, 2, 3;
0, 1, 3, 6, 6;
0, 0, 2, 9, 12, 16;
0, 0, 2, 12, 18, 32, 39;
0, 0, 1, 9, 24, 48, 78, 103;
0, 0, 1, 9, 30, 64, 117, 206, 263;
0, 0, 0, 6, 24, 80, 156, 309, 526, 690;
0, 0, 0, 6, 24, 96, 195, 412, 789, 1380, 1791;
0, 0, 0, 3, 18, 80, 234, 515, 1052, 2070, 3582, 4693;
0, 0, 0, 3, 18, 80, 273, 618, 1315, 2760, 5373, 9386, 12247;
0, 0, 0, 0, 12, 64, 234, 721, 1578, 3450, 7164, 14079, 24494, 32073;
...

Cf. A168316, A168317

Triangle read by rows, M*Q. M = A168316, Q = an infinite lower triangular matrix
with A168317 prefaced with a 1; (1, 1, 1, 3, 6, 16, 39, 103,...) as the right
diagonal and the rest zeros.

A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 7, 8, 12, 14, 19, 21, 27, 30, 37, 40, 48, 52, 61, 65, 75, 80, 91, 96, 108, 114, 127, 133, 147, 154, 169, 176, 192, 200, 217, 225, 243, 252, 271, 280, 300, 310, 331, 341, 363, 374, 397, 408, 432, 444, 469, 481, 507, 520, 547, 560, 588, 602, 631

Offset: 0

Paul Curtz, May 20 2007

nonn

Interleaving of A077043 and A006578.
First differences are in A124072.
If the values of the second, fourth, sixth, ... column are replaced by the corresponding negative values, the antidiagonal sums of the resulting triangular array are 0, 0, 1, 1, -1, -2, -1, -2, -6, -8, -7, -9, ... .
Row sums of triangle A168316 = (1, 1, 3, 4, 7, 8, 12, ...). - Gary W. Adamson, Nov 22 2009

			First seven rows of T are
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3, 2;
  0, 1, 4, 2, 3;
  0, 1, 5, 2, 4, 3;
  0, 1, 6, 2, 5, 3, 4;.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006578, A077043, A124072, A168316.

m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jul 16 2007

A129819:= func< n | Floor(((n-1)*(3*n+1) +(2*n+5)*((n+1) mod 2))/16) >;
[A129819(n): n in [0..70]]; // G. C. Greubel, Sep 19 2024

CoefficientList[Series[x^2*(1+x^2+x^3)/((1-x)*(1-x^2)*(1-x^4)), {x, 0, 70}], x] (* G. C. Greubel, Sep 19 2024 *)

{vector(59, n, (n-2+n%2)*(n+n%2)/8+floor((n-2-n%2)^2/16))} \\ Klaus Brockhaus, Jul 16 2007

def A129819(n): return ((n-1)*(3*n+1) + (2*n+5)*((n+1)%2))//16
[A129819(n) for n in range(71)] # G. C. Greubel, Sep 19 2024

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n > 6, with a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 4, a(6) = 7.
G.f.: x^2*(1+x^2+x^3)/((1-x)^3*(1+x)^2*(1+x^2)).
a(n) = (3/16)*(n+2)*(n+1) - (5/8)*(n+1) + 7/32 + (3/32)*(-1)^n + (1/16)*(n+1)*(-1)^n - (1/8)*cos(n*Pi/2) + (1/8)*sin(n*Pi/2). - Richard Choulet, Nov 27 2008

Edited and extended by Klaus Brockhaus, Jul 16 2007

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A168317 Eigensequence of triangle A168316.

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A168318 Triangle read by rows, A168316 * its diagonalized eigensequence, A168317.

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A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

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