A133080 -id:A133080 - cp's OEIS frontend

A133114 A000012 * A007318 * A133080.

1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 3, 2, 7, 3, 1, 3, 3, 13, 7, 5, 1, 4, 3, 22, 13, 16, 5, 1, 4, 4, 34, 22, 40, 16, 7, 1, 5, 4, 50, 34, 86, 40, 29, 7, 1, 5, 5, 70, 50, 166, 86, 91, 29, 9, 1

Offset: 1

Gary W. Adamson, Sep 14 2007

Row sums = 2^n

			First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
2, 2, 3, 1;
3, 2, 7, 3, 1;
3, 3, 13, 7, 5, 1;
4, 3, 22, 13, 16, 5, 1;
4, 4, 34, 22, 40, 16, 7, 1;
...

Cf. A000012, A133080, A007318.

A000012(signed) * A007318 * A133080 as infinite lower triangular matrices, where A000012(signed) = [1; -1,1; 1,-1,1; -1,1,-1,1;...).

A133571 (A007318 * A133080 + A133080 * A007318) - A007318.

Original entry on oeis.org

1, 3, 1, 3, 2, 1, 5, 5, 5, 1, 5, 4, 10, 4, 1, 7, 9, 26, 14, 7, 1, 7, 6, 35, 20, 21, 6, 1, 9, 13, 71, 55, 71, 27, 9, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 11, 17, 148, 140, 322, 182, 14844, 11, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A133572: (1, 4, 6, 16, 24, 64, 96, 256, ...).

			First few rows of the triangle:
  1;
  3,  1;
  3,  2,  1;
  5,  5,  5,  1;
  5,  4, 10,  4,  1;
  7,  9, 26, 14,  7,  1;
  7,  6, 35, 20, 21,  6,  1;
  9, 13, 71, 55, 71, 27,  9,  1;
  ...

Cf. A133080, A133572.

(A007318 * A133080 + A133080 * A007318) - A007318, as infinite lower triangular matrices, where A133080 is an interpolation operator.

A133601 A007318 * (A097806 + A133080 - I), I = Identity matrix.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 6, 5, 1, 9, 10, 14, 5, 1, 11, 15, 30, 15, 7, 1, 13, 21, 55, 35, 27, 7, 1, 15, 28, 91, 70, 77, 28, 9, 1, 17, 36, 140, 126, 182, 84, 44, 9, 1, 19, 45, 204, 210, 378, 210, 156, 45, 11, 1

Offset: 0

Gary W. Adamson, Sep 18 2007

			First few rows of the triangle are:
1;
3, 1;
5, 3, 1;
7, 6, 5, 1;
9, 10, 14, 5, 1;
11, 15, 30, 15, 7, 1;
13, 21, 55, 35, 27, 7, 1;
15, 28, 91, 70, 77, 28, 9, 1;
...

Cf. A097806, A133080, A052549 (row sums).

A133601aux := proc(n,k)
    if n <> k then
        A097806(n,k)+A133080(n,k) ;
    else
        A097806(n,k)+A133080(n,k)-1 ;
    end if;
end proc:
A133601 := proc(n,k)
    add( A007318(n,j)*A133601aux(j+1,k+1),j=k..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A007318 * (A097806 + A133080 - I), I = Identity matrix. Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (2,1,2,1,2,...) in the subdiagonal; and the rest zeros.

A133602 The matrix-vector product A133080 * A000108.

Original entry on oeis.org

1, 2, 2, 7, 14, 56, 132, 561, 1430, 6292, 16796, 75582, 208012, 950912, 2674440, 12369285, 35357670, 165002460, 477638700, 2244901890, 6564120420, 31030387440, 91482563640, 434542177290, 1289904147324, 6151850548776

Offset: 0

Gary W. Adamson, Sep 18 2007

A133603 is a companion sequence.

			a(4) = C(4) = 14.
a(5) = 56 = C(5) + C(4) = 42 + 14.

Cf. A097806, A133080, A000108, A133603.

from sympy import catalan
def a005807(n): return catalan(n) + catalan(n + 1)
def a048990(n): return catalan(2*n)
l=[1, 2]
for n in range(2, 31): l+=[a048990(n//2) if n%2==0 else a005807(n - 1)]
print(l) # Indranil Ghosh, Jul 15 2017

A133080 * A000108, where A133080 = an infinite lower triangular matrix and A000108 = the Catalan sequence as a vector.
a(2n) = A048990(n).
a(2n+1) = A005807(2n).
Conjecture: n*(n-1)*(n-3)*(3*n-4)*a(n) -8*(n-1)*(2*n-5)*a(n-1) -4*(n-2)*(3*n-1)*(2*n-5)*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jun 20 2015

A133804 Triangle read by rows: A007318 * A133080 * A133566.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 14, 10, 5, 1, 6, 25, 20, 16, 6, 1, 7, 41, 35, 41, 21, 7, 1, 8, 63, 56, 91, 56, 29, 8, 1, 9, 92, 84, 182, 126, 92, 36, 9, 1, 10, 129, 120, 336, 252, 246, 120, 46, 10, 1, 11, 175, 165, 582, 462, 582, 330, 175, 55, 11, 1, 12, 231, 220, 957, 792, 1254, 792, 550, 220, 67, 12, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133124: (1, 3, 7, 16, 35, 74, 153, ...).

A133805 = binomial transform of (A133566 * A133080).

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  7,  4,  1;
  5, 14, 10,  5,  1;
  6, 25, 20, 16,  6,  1;
  7, 41, 35, 41, 21,  7,  1;
  ...

Cf. A133080, A133124, A133566.

Binomial transform of (A133080 * A133566), where A133080 * A133566 = an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (0,1,0,1,0,...) in the subsubdiagonal.

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

A133938 A007318 * (A129686 + A133080 - I), where I is the identity matrix.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 4, 4, 1, 11, 8, 11, 4, 1, 16, 15, 25, 11, 6, 1, 22, 26, 50, 26, 22, 6, 1, 29, 42, 91, 56, 63, 22, 8, 1, 37, 64, 154, 112, 154, 64, 37, 8, 1, 46, 93, 24, 210, 336, 162, 129, 37, 10, 1

Offset: 1

Gary W. Adamson, Sep 29 2007

Left column = A000124: (1, 2, 4, 7, 11, ...).

Row sums = A133124: (1, 3, 7, 16, 35, 74, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  2,  1;
   7,  4,  4,  1;
  11,  8, 11,  4,  1;
  16, 15, 25, 11,  6,  1;
  22, 26, 50, 26, 22,  6,  1;
  ...

Cf. A000124, A133124, A129686, A133080.

Binomial transform of matrix M, where M = a tridiagonal matrix with (1,1,1,...) in the main diagonal, (1,0,1,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal. M = (A129686 + A133080 - I), I = Identity matrix.

A133088 A007318^(-1) * A133080.

Original entry on oeis.org

1, 0, 1, -1, -2, 1, 2, 3, -2, 1, -3, -4, 2, -4, 1, 4, 5, 0, 10, -4, 1, -5, -6, -5, -20, 9, -6, 1, 6, 7, 14, 35, -14, 21, -6, 1, -7, -8, -28, -56, 14, -56, 20, -8, 1, 8, 9, 48, 84, 0, 126, -48, 36, -8, 1, -9, -10, -75, -120, -42, -252, 90, -120, 35, -10, 1

Offset: 0

Gary W. Adamson, Sep 08 2007

Row sums give A123344.

Inverse binomial transform of A133080.

			First few rows of the triangle:
   1;
   0,   1;
  -1,  -2,   1;
   2,   3,  -2,   1;
  -3,  -4,   2,  -4,   1;
   4,   5,   0,  10,  -4,   1;
  -5,  -6,  -5, -20,   9,  -6,   1;
  ...

Cf. A133080, A007318.

tabl(nn) = {t007318 = matrix(nn, nn, n, k, binomial(n-1, k-1)); t133080 = matrix(nn, nn, n, k, if (k==n, 1, if (k == (n-1), 1 - (n % 2), 0))); t133088 = t007318^(-1)*t133080; for (n = 1, nn, for (k = 1, n, print1(t133088[n, k], ", ");); print(););} \\ Michel Marcus, Feb 13 2014

Typo corrected by Travis Hoppe, Apr 24 2008

One term's sign corrected by Michel Marcus, Feb 13 2014

A093178 If n is even then 1, otherwise n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1, 85

Offset: 0

Michael Somos, Mar 27 2004

Continued fraction expansion for tan(1).
1 followed by run lengths of A062557 = 2n-1 1's followed by a 2. - Jeremy Gardiner, Aug 12 2012
Greatest common divisor of n and (n+1) mod 2. - Bruno Berselli, Mar 07 2017

			1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...))))
G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 5*x^5 + x^6 + 7*x^7 + x^8 + 9*x^9 + x^10 + ...

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Simon Plouffe, A Search for a mathematical expression for mass ratios using a large database. page 3.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Equals |A009001(n)|.

Cf. A133080, A049471 (decimal expansion), A009001, A161738, A062557, A124625.

A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* Harvey P. Dale, Nov 23 2011 *)

PARI
```
{a(n) = if( n%2, n, 1)};
```

G.f.: (1+x-x^2+x^3)/(1-x^2)^2.
a(n) = (-1)^n * a(-n) for all n in Z.
a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - Ralf Stephan, Dec 02 2004
a(n) = n^n mod (n+1) for n > 0. - Amarnath Murthy, Apr 18 2004
Satisfies a(0) = 1, a(n+1) = a(n) + n if a(n) < n else a(n+1) = a(n)/n. - Amarnath Murthy, Oct 29 2002
a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - Paul Barry, Apr 09 2003
a(n) = binomial(n, 2*floor(n/2)). - Paul Barry, Dec 28 2006
Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
a(n) = denom(b(n+2)/b(n+1)) with b(n) = product((2*n-3-2*k), k=0..floor(n/2-1)). - Johannes W. Meijer, Jun 18 2009
a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - Wesley Ivan Hurt, Oct 19 2013
a(n) = n^(n mod 2). - Wesley Ivan Hurt, Apr 16 2014

A084221 a(n+2) = 4*a(n), with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 4, 12, 16, 48, 64, 192, 256, 768, 1024, 3072, 4096, 12288, 16384, 49152, 65536, 196608, 262144, 786432, 1048576, 3145728, 4194304, 12582912, 16777216, 50331648, 67108864, 201326592, 268435456, 805306368, 1073741824, 3221225472, 4294967296, 12884901888

Offset: 0

Paul Barry, May 21 2003

Binomial transform is A060925. Binomial transform of A084222.
Sequences with similar recurrence rules: A016116 (multiplier 2), A038754 (multiplier 3), A133632 (multiplier 5). See A133632 for general formulas. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * A000079. A122756 is a companion sequence. - Gary W. Adamson, Sep 19 2007

			Binary...............Decimal
1..........................1
11.........................3
100........................4
1100......................12
10000.....................16
110000....................48
1000000...................64
11000000.................192
100000000................256
1100000000...............768
10000000000.............1024
110000000000............3072, etc. - _Philippe Deléham_, Mar 21 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hester Graves, The Minimal Euclidean Function on the Gaussian Integers, arXiv:1802.08281 [math.NT], 2018. See Definition 2.3 p.3.
Index entries for linear recurrences with constant coefficients, signature (0,4)

For partial sums see A133628. Partial sums for other multipliers p: A027383(p=2), A087503(p=3), A133629(p=5).
Other related sequences: A132666, A132667, A132668, A132669.
Cf. A000302, A133080, A133087, A164346.

[(5*2^n-(-2)^n)/4: n in [0..40]]; // Vincenzo Librandi, Aug 13 2011

CoefficientList[Series[(-3*x - 1)/(4*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

a(n)=([0,1; 4,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = (5*2^n-(-2)^n)/4.
G.f.: (1+3*x)/((1-2*x)(1+2*x)).
E.g.f.: (5*exp(2*x) - exp(-2*x))/4.
a(n) = A133628(n) - A133628(n-1) for n>1. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * [1, 2, 4, 8, ...]. Row sums of triangle A133087. - Gary W. Adamson, Sep 08 2007
a(n+1)-2a(n) = A000079 signed. a(n)+a(n+2)=5*a(n). First differences give A135520. - Paul Curtz, Apr 22 2008
a(n) = A074323(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = Sum_{k=0..n+1} A181650(n+1,k)*2^k. - Philippe Deléham, Nov 19 2011
a(2*n) = A000302(n); a(2*n+1) = A164346(n). - Philippe Deléham, Mar 21 2014

Edited by N. J. A. Sloane, Dec 14 2007

A133566 Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Usually regarded as a square matrix T when combined with other matrices and column vectors.
Then T * V, where V = any sequence regarded as a column vector with offset 1 is a new sequence S [called an interpolation transform] given by S(2n) = V(2n), S(2n-1) = V(2n) + V(2n-1). Example: If T * [1,2,3,...], S = [1, 2, 5, 4, 9, 6, 13, 8, 17, ...) = A114752. A133080 is identical to A133566 except that the subdiagonal = (1,0,1,0,...). A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, 9, 19, ...).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,-1,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2007

			First few rows of the triangle:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1;
  ...

Cf. A133080, A114752, A114753, A084938, A040001.

A133566 := proc(n,k)
    if n = k then
        1;
    elif  k=n-1 and type(n,odd) then
        1;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

T[n_, k_] := Which[n == k, 1, k == n - 1 && OddQ[n], 1, True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 24 2023 *)

Odd rows: (n-2) zeros followed by 1, 1. Even rows: (n-1) zeros followed by 1.
Sum_{k=0..n} T(n,k) = A040001(n). - Philippe Deléham, Dec 15 2007
G.f.: (-1-x*y-x^2*y)*x*y/((-1+x*y)*(1+x*y)). - R. J. Mathar, Aug 11 2015

Entry revised by N. J. A. Sloane, Jun 20 2015

cp's OEIS Frontend

A133114 A000012 * A007318 * A133080.

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A133571 (A007318 * A133080 + A133080 * A007318) - A007318.

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A133601 A007318 * (A097806 + A133080 - I), I = Identity matrix.

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A133602 The matrix-vector product A133080 * A000108.

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A133804 Triangle read by rows: A007318 * A133080 * A133566.

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A133938 A007318 * (A129686 + A133080 - I), where I is the identity matrix.

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A133088 A007318^(-1) * A133080.

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A093178 If n is even then 1, otherwise n.

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A084221 a(n+2) = 4*a(n), with a(0)=1, a(1)=3.

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