A084221 -id:A084221 - cp's OEIS frontend

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

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

A087440 Expansion of (1-2x-3x^2)/((1-2x)(1-4x)).

Original entry on oeis.org

1, 4, 13, 46, 172, 664, 2608, 10336, 41152, 164224, 656128, 2622976, 10488832, 41949184, 167784448, 671113216, 2684403712, 10737516544, 42949869568, 171799085056, 687195553792, 2748780642304, 10995119423488, 43980471402496

Offset: 0

Paul Barry, Sep 03 2003

Binomial transform is A087439. Second binomial transform of A084221 (with extra leading 1).

Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000217, A084221, A087439.

CoefficientList[Series[(1-2x-3x^2)/((1-2x)(1-4x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{6,-8},{4,13},30]] (* Harvey P. Dale, Jan 18 2012 *)

a(n) = 5*4^n/8 + 3*2^n/4 - 3*0^n/8.
a(n) = 6*a(n-1) - 8*a(n-2), n>2. - Harvey P. Dale, Jan 18 2012
a(n) = A000217(2^n) + floor(A000217(2^(n-1))). - J. M. Bergot, May 03 2018

A133087 A133080 * A007318.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 5, 4, 1, 1, 4, 6, 4, 1, 2, 9, 16, 14, 6, 1, 1, 6, 15, 20, 15, 6, 1, 2, 13, 36, 55, 50, 27, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1

Offset: 0

Gary W. Adamson, Sep 08 2007

Row sums = A084221: (1, 3, 4, 12, 16, 48, 64, 192, ...).

Subtriangle of (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  5,  4,  1;
  1,  4,  6,  4,  1;
  2,  9, 16, 14,  6,  1;
  1,  6, 15, 20, 15,  6,  1;
  2, 13, 36, 55, 50, 27,  8,  1;
  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...
Triangle (0, 2, -3/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  1,  2,  1;
  0,  2,  5,  4,  1;
  0,  1,  4,  6,  4,  1;
  0,  2,  9, 16, 14,  6,  1;
  0,  1,  6, 15, 20, 15,  6,  1;
  0,  2, 13, 36, 55, 50, 27,  8,  1;
  0,  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A133080, A084221.

Columns: A000007, A114752, A133092.

CoefficientList[CoefficientList[Series[(1 + 2*x + y*x)/(1 - (1 + y)^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 21 2017 *)

A133080 * A007318 as infinite lower triangular matrices.
G.f.: (1+2*x+y*x)/(1-(1+y)^2*x^2). - Philippe Deléham, Mar 03 2012
T(n,k) = T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 1. - Philippe Deléham, Mar 03 2012
Sum_{k=0..n} T(n,k)*x^k = A059841(n), A019590(n+1), A000034(n), A084221(n), A133125(n) for x = -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Mar 03 2012

A181650 Inverse of number triangle A070909.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1

Offset: 0

Paul Barry, Nov 03 2010

Generalized (conditional) Riordan array with k-th column generated by x^k*(1-x-x^2) if k is even, x^k otherwise.
Triangle T(n,k), read by rows, given by (-1,2,-1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (1,0,-1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011
Double Riordan array (1 - x - x^2; x/(1 - x - x^2), x*(1 - x - x^2)) as defined in Davenport et al. - Peter Bala, Aug 15 2021

			Triangle begins
   1,
  -1,  1,
  -1,  0,  1,
   0,  0, -1,  1,
   0,  0, -1,  0,  1,
   0,  0,  0,  0, -1,  1,
   0,  0,  0,  0, -1,  0,  1,
   0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0,  0,  0, -1,  0,  1,
   0,  0,  0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  1
Production matrix begins
  -1,  1,
  -2,  1,  1,
  -1,  1, -1,  1,
  -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1,
  -1,  1, -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1, -1,  1,
  -1,  1, -1,  1, -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1, -1,  1, -1,  1

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A084221, A084938.

G.f.: (1+(y-1)*x-x^2)/((1-y*x)*(1+y*x)). - Philippe Deléham, Nov 19 2011

A137447 a(n) = 4*a(n-4) for n>3, a(0)=-1, a(1)=-4, a(2)=2, a(3)=12.

Original entry on oeis.org

-1, -4, 2, 12, -4, -16, 8, 48, -16, -64, 32, 192, -64, -256, 128, 768, -256, -1024, 512, 3072, -1024, -4096, 2048, 12288, -4096, -16384, 8192, 49152, -16384, -65536, 32768, 196608, -65536, -262144, 131072, 786432, -262144, -1048576, 524288, 3145728, -1048576

Offset: 0

Paul Curtz, Apr 18 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4).

Cf. A084221, A102561, A122803.

&cat[[-(-2)^n,2^n-5*(-2)^n]: n in [0..20]];  // Bruno Berselli, Nov 02 2011

LinearRecurrence[{0,0,0,4},{-1,-4,2,12},50] (* or *) CoefficientList[ Series[(1+4x-2x^2-12x^3)/(4x^4-1),{x,0,50}],x] (* Harvey P. Dale, Jun 27 2011 *)

def A137447(n): return 2^(n//2)*(-1)^(n//2+1) if n%2==0 else 2^((n-1)//2)*(1 - 5*(-1)^((n-1)//2))
[A137447(n) for n in range(51)] # G. C. Greubel, Sep 15 2023

G.f.: (1+4*x-2*x^2-12*x^3)/(4*x^4-1). - Harvey P. Dale, Jun 27 2011
From Bruno Berselli, Nov 02 2011: (Start)
a(n) = (1-(-1)^n-2*(3-2*(-1)^n)*(-1)^floor(n/2))*2^(floor(n/2)-1).
a(2n) = -A122803(n).
a(2n+1) = (-1)^(n+1)*A084221(n+2). (End)
E.g.f.: (1/sqrt(2))*( sinh(sqrt(2)*x) - 5*sin(sqrt(2)*x) - sqrt(2)*cos(sqrt(2)*x) ). - G. C. Greubel, Sep 15 2023

More terms from Harvey P. Dale, Jun 27 2011

A240525 a(n) = 2^(n-2)*(2^(n+4)-(-1)^n+5).

Original entry on oeis.org

5, 19, 68, 268, 1040, 4144, 16448, 65728, 262400, 1049344, 4195328, 16780288, 67112960, 268447744, 1073758208, 4295016448, 17179934720, 68719673344, 274878169088, 1099512414208, 4398047559680, 17592189190144, 70368748371968, 281474989293568, 1125899923619840

Offset: 0

Vincenzo Librandi, Apr 07 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Cf. A000302, A084221.

[2^(n-2)*(2^(n+4)-(-1)^n+5): n in [0..25]];

I:=[5,19,68]; [n le 3 select I[n] else 4*Self(n-1)+4*Self(n-2)-16*Self(n-3): n in [1..30]];

CoefficientList[Series[(5 - x - 28 x^2)/(1 - 4 x - 4 x^2 + 16 x^3), {x, 0, 33}], x]

a(n)=(2^(n+4)-(-1)^n+5)<<(n-2) \\ Charles R Greathouse IV, Aug 26 2014

G.f.: (5-x-28*x^2)/(1-4*x-4*x^2+16*x^3).
a(n) = 4*a(n-1) + 4*a(n-2)- 16*a(n-3) with n>2, a(0)=5, a(1)=19, a(2)=68.
a(n) = (5*2^n-(-2)^n)/4+4^(n+1) = A084221(n)+A000302(n+1).

cp's OEIS Frontend

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

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A087440 Expansion of (1-2x-3x^2)/((1-2x)(1-4x)).

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A133087 A133080 * A007318.

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A181650 Inverse of number triangle A070909.

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A137447 a(n) = 4*a(n-4) for n>3, a(0)=-1, a(1)=-4, a(2)=2, a(3)=12.

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A240525 a(n) = 2^(n-2)*(2^(n+4)-(-1)^n+5).

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