cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A203835 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having equal diagonal elements or equal antidiagonal elements.

Original entry on oeis.org

45, 225, 225, 1125, 1971, 1125, 5625, 17289, 17289, 5625, 28125, 151659, 270333, 151659, 28125, 140625, 1330353, 4238721, 4238721, 1330353, 140625, 703125, 11669859, 66490965, 119606211, 66490965, 11669859, 703125, 3515625, 102368025
Offset: 1

Views

Author

R. H. Hardin Jan 06 2012

Keywords

Comments

Table starts
......45.......225.........1125...........5625.............28125
.....225......1971........17289.........151659...........1330353
....1125.....17289.......270333........4238721..........66490965
....5625....151659......4238721......119606211........3383285769
...28125...1330353.....66490965.....3383285769......173185290765
..140625..11669859...1043088057....95763046491.....8882824128417
..703125.102368025..16363800045..2710984443345...455911325162757
.3515625.897972507.256713156657.76749227497395.23404859123410809

Examples

			Some solutions for n=4 k=3
..0..2..0..1....2..1..0..0....0..1..2..0....0..0..2..0....2..1..0..2
..2..0..1..0....1..1..1..0....1..2..2..2....0..2..2..2....2..2..1..0
..0..2..0..1....2..1..2..1....0..1..2..2....1..0..2..2....1..2..2..1
..2..0..0..0....2..2..2..2....0..0..1..2....2..1..0..2....0..1..2..2
..2..2..0..1....1..2..1..2....2..0..0..1....1..2..1..0....2..0..1..2

Links

Crossrefs

Column 1 is A189274(n+2)

Formula

Empirical for column k:
k=1: a(n) = 9*5^n
k=2: a(n) = 9*a(n-1) -2*a(n-2)
k=3: a(n) = 19*a(n-1) -54*a(n-2) +32*a(n-3)
k=4: a(n) = 31*a(n-1) -24*a(n-2) -1612*a(n-3) +3816*a(n-4) +1152*a(n-5) -2784*a(n-6) +256*a(n-7)
k=5: (order 12 recurrence)
k=6: (order 28 recurrence)
k=7: (order 54 recurrence)

A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.

Original entry on oeis.org

1, 3, 2, 6, 7, 2, 12, 20, 11, 2, 24, 52, 42, 15, 2, 48, 128, 136, 72, 19, 2, 96, 304, 400, 280, 110, 23, 2, 192, 704, 1104, 960, 500, 156, 27, 2, 384, 1600, 2912, 3024, 1960, 812, 210, 31, 2, 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2, 1536, 7936
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Examples

			First five rows:
   1;
   3,  2;
   6,  7,  2;
  12, 20, 11,  2;
  24, 52, 42, 15,  2;

Crossrefs

Cf. A207635.
Cf. A084938, A003945, A040000.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207635 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207636 *)

Formula

u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 3, T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x+y*x)/(1-2*x-y*x).
Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)

A270567 Expansion of g.f. (1+4*x)/(1-5*x).

Original entry on oeis.org

1, 9, 45, 225, 1125, 5625, 28125, 140625, 703125, 3515625, 17578125, 87890625, 439453125, 2197265625, 10986328125, 54931640625, 274658203125, 1373291015625, 6866455078125, 34332275390625, 171661376953125, 858306884765625, 4291534423828125, 21457672119140625, 107288360595703125
Offset: 0

Views

Author

Colin Barker, Mar 19 2016

Keywords

Comments

Partial sums are 1, 10, 55, 280, 1405, 7030, ...
Apparently a duplicate of A189274. - R. J. Mathar, May 13 2016

Links

Crossrefs

Cf. A000351 (powers of 5), A128625 (1+3*x)/(1-5*x), A189274.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 4 x)/(1 - 5 x), {x, 0, 23}], x] (* Michael De Vlieger, Mar 19 2016 *)
  • PARI
    Vec((1+4*x)/(1-5*x) + O(x^30))

Formula

G.f.: (1+4*x)/(1-5*x).
a(n) = 5*a(n-1) for n>1.
a(n) = 9*5^(n-1) for n>0.
E.g.f.: (9*exp(5*x) - 4)/5. - Elmo R. Oliveira, Mar 25 2025

A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.

Original entry on oeis.org

1, 15, 117, 891, 6777, 51543, 392013, 2981475, 22675761, 172461663, 1311666021, 9975943179, 75872547369, 577052549415, 4388802753213, 33379264377459, 253867706760033, 1930803860947887, 14684827767302997, 111686210555580315, 849435201142733529, 6460422977475127287
Offset: 0

Views

Author

George Strand Vajagich, Mar 01 2020

Keywords

Examples

			For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 2 to get 102 add 15 to get 117.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 2 to get 774 add 177 to get 891.

Links

Crossrefs

Cf. A332936 (number of blue nodes).
Similar sequences with a cycle size 3..6 are: A007483, A048876, A189274(n+1), A054490.

Programs

  • PARI
    Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ Colin Barker, Mar 03 2020
  • Python
    g=1
b=7
sg=0
sb=0
bl=[]
gl=[]
for int in range(1,20):
  sg=g*1+b*2
  sb=b*7+g*2
  g=sg
  b=sb
  gl.append(g)
  bl.append(b)
print(gl)

Formula

a(n) = a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) + 7*b(n-1) with a(0) = 1 and b(0) = 7 where b(n) = A332936(n).
From Colin Barker, Mar 03 2020: (Start)
G.f.: (1 + 7*x) / (1 - 8*x + 3*x^2).
a(n) = 8*a(n-1) - 3*a(n-2) for n>1.
(End)
From Stefano Spezia, Mar 03 2020: (Start)
a(n) = ((4 - sqrt(13))^n*(-11 + sqrt(13)) + (4 + sqrt(13))^n*(11 + sqrt(13)))/(2*sqrt(13)).
E.g.f.: exp(4*x)*cosh(sqrt(13)*x) + (11*exp(4*x)*sinh(sqrt(13)*x))/sqrt(13).
(End)

Extensions

a(14)-a(21) from Stefano Spezia, Mar 03 2020
Typo in a(14) fixed by Colin Barker, Apr 26 2020
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.