A131023 First subdiagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

1, 2, 3, 4, 5, 7, 14, 37, 101, 256, 593, 1267, 2534, 4825, 8921, 16384, 30581, 58975, 117950, 242461, 504605, 1048576, 2156201, 4371451, 8742902, 17308657, 34085873, 67108864, 132623405, 263652487, 527304974, 1059392917, 2133134741

Offset: 1

Author

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Keywords

• nonn
• easy

Comments

Also first differences of main diagonal A129339.

Examples

			For first seven rows of T see A131022 or A129339.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-9,6).

Crossrefs

Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

Programs

Magma

m:=34; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n+1, n]: n in [1..m-1] ];

PARI

{m=33; v=concat([1, 2, 3, 4],vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v}

Formula

a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).

G.f.: x*(1-3*x+2*x^2+x^3)/((1-2*x)*(1-3*x+3*x^2)).

a(n) = A057681(n-1) + 2^(n-2), a(1) = 1. - Bruno Berselli, Feb 17 2011