A306713
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).
Original entry on oeis.org
1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, 0, ...
3, 1, 1, 0, 0, 0, 0, 0, 0, ...
5, 1, 1, 1, 0, 0, 0, 0, 0, ...
8, 2, 0, 1, 1, 0, 0, 0, 0, ...
13, 2, 1, 0, 1, 1, 0, 0, 0, ...
21, 3, 2, 0, 0, 1, 1, 0, 0, ...
34, 4, 1, 1, 0, 0, 1, 1, 0, ...
55, 5, 1, 2, 0, 0, 0, 1, 1, ...
-
T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)
A306752
a(n) = Sum_{k=0..n} binomial(k, 8*(n-k)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6436, 12871, 24312, 43776, 75736, 126940, 208336, 340120, 564928, 980629, 1817047, 3605252, 7531836, 16146326, 34716826, 73737316, 153430156, 311652271, 617594122, 1195477615, 2266064352, 4221317464
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..3528
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1,1).
-
a[n_] := Sum[Binomial[k, 8*(n-k)], {k, 0, n}]; Array[a, 38, 0] (* Amiram Eldar, Jun 21 2021 *)
-
{a(n) = sum(k=0, n, binomial(k, 8*(n-k)))}
-
N=66; x='x+O('x^N); Vec((1-x)^7/((1-x)^8-x^9))
A127843
a(1) = 1, a(2) = ... = a(9) = 0, a(n) = a(n-9)+a(n-8) for n>9.
Original entry on oeis.org
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 2, 9
Offset: 1
Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007
- S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,1).
-
a:=[1,0,0,0,0,0,0,0,0];; for n in [10..90] do a[n]:=a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 07 2018
-
LinearRecurrence[{0,0,0,0,0,0,0,1,1},{1,0,0,0,0,0,0,0,0},120] (* Harvey P. Dale, Jun 15 2017 *)
CoefficientList[Series[(1-x)*(1+x)*(1+x^2)*(1+x^4) / (1-x^8-x^9), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *)
-
Vec(x*(1-x)*(1+x)*(1+x^2)*(1+x^4)/(1-x^8-x^9) + O(x^100)) \\ Colin Barker, May 30 2016
A376648
a(n) = Sum_{k=0..floor(n/4)} binomial(floor(k/2),n-4*k).
Original entry on oeis.org
1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 2, 4, 6, 4, 2, 5, 10, 10, 6, 6, 10, 10, 6, 7, 15, 20, 16, 12, 16, 20, 16, 13, 22, 35, 36, 28, 28, 36, 36, 29, 35, 57, 71, 64, 56, 64, 72, 65, 64, 92, 128, 135
Offset: 0
-
a(n) = sum(k=0, n\4, binomial(k\2, n-4*k));
-
my(N=80, x='x+O('x^N)); Vec((1+x^4)/(1-x^8-x^9))
Showing 1-4 of 4 results.
Comments