A128625 -id:A128625 - cp's OEIS frontend

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

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

Clark Kimberling, Feb 24 2012

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

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

Cf. A207635.

Cf. A084938, A003945, A040000.

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 *)

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)

A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

Original entry on oeis.org

5, 18, 35, 50, 180, 315, 350, 500, 1800, 3150, 3500, 5000, 18000, 31500, 35000, 50000, 180000, 315000, 350000, 500000, 1800000, 3150000, 3500000, 5000000, 18000000, 31500000, 35000000, 50000000, 180000000, 315000000, 350000000, 500000000, 1800000000

Offset: 1

M. F. Hasler, Feb 09 2018

The only primitive terms of the sequence, i.e., not equal to 10 times a smaller term, are 5, 18, 35 and 315.
For base 2, 3, 4 and 5, the corresponding sequences are less interesting: b = 2 yields powers of 2, A000079; b = 3 yields 4 times powers of 3, A003946 \ {1}; b = 4 yields {2, 6}*{4^k, k>=0} = A122756 = 2*A084221; b = 5 yields 8*{5^k, k>=0} = A128625 \ {1}.
See A298976 for base-6 complementary numbers. Base 7 yields {12, 120}*{7^k, k>=0}, cf. A298977. The linked web page (in French) gives also examples for base-100 complementary numbers, e.g., 198 = (100 - 1)*(100 - 98), 1680 = (100 - 16)*(100 - 80), ..., and for base-1000 complementary numbers.

			5 = (10-5), therefore 5 is in the sequence.
18 = (10-1)*(10-8), therefore 18 is in the sequence.
35 = (10-3)*(10-5), therefore 35 is in the sequence.
315 = (10-3)*(10-1)*(10-5), therefore 315 is in the sequence.
If x is in the sequence, then 10*x = concat(x,0) = x*(10-0) is in the sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
G. Villemin, Nombres complémentés (in French).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,10).

Cf. A298976, A298977.

LinearRecurrence[{0,0,0,10},{5,18,35,50,180,315},40] (* Harvey P. Dale, Mar 02 2024 *)

is(n,b=10)={n==prod(i=1,#n=digits(n,b),b-n[i])}

a(n)=if(n>6,a((n-3)%4+3)*10^((n-3)\4),[5,18,35,50,180,315][n])

Vec(x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4) + O(x^60)) \\ Colin Barker, Feb 09 2018

a(n+4) = 10 a(n) for all n >= 3.

G.f.: x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4). - Colin Barker, Feb 09 2018

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

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

Colin Barker, Mar 19 2016

Partial sums are 1, 10, 55, 280, 1405, 7030, ...

Apparently a duplicate of A189274. - R. J. Mathar, May 13 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5).

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

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))
```

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

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

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A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

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

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cp's OEIS Frontend

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

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Formula

A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

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Author

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Comments

Examples

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PARI

PARI

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Formula

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

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PARI

Formula

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