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.

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A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

Original entry on oeis.org

1, 3, 7, 11, 23, 59, 103, 139, 279, 827, 1895, 2955, 5655, 14395, 24679, 32907, 65815, 197435, 460647, 723851, 1512983, 3881019, 6774887, 9142411, 18219287, 54002491, 123733863, 192940939, 369104407, 939538491, 1610637415, 2147516555
Offset: 0

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Author

Paul D. Hanna, Oct 30 2004

Keywords

Comments

Equals the XOR BINOMIAL transform of A099901. Also, equals the main diagonal of the XOR difference triangle A099900, in which the central terms of the rows form the powers of 2.
Bisection of A101624. - Paul Barry, May 10 2005

Links

Crossrefs

Cf. A099884, A099900, A099901.

Programs

  • Maple
    a:= n -> add((binomial(n-k+floor(k/2),floor(k/2)) mod 2)*2^k, k=0..n):
map(a, [$0..100]); # Robert Israel, Jan 24 2016
  • PARI
    {a(n)=local(B);B=0;for(k=0,n,B=bitxor(B,binomial(n-k+k\2,k\2)%2*2^k));B}
  • PARI
    a(n)=sum(k=0,n,binomial(n-k+k\2,k\2)%2*2^k)
  • Python
    def A099902(n): return sum(int(not ~((n<<1)-k)&k)<Chai Wah Wu, Jul 30 2025

Formula

a(n) = SumXOR_{k=0..n} (binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k for n >= 0.
a(n) = SumXOR_{i=0..n} (C(n, i) mod 2)*A099901(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j) mod 2 = A047999(i, j).
a(n) = Sum_{k=0..n} A047999(n-k+floor(k/2), floor(k/2)) * 2^k.
From Paul Barry, May 10 2005: (Start)
a(n) = Sum_{k=0..2n} (binomial(k, 2n-k) mod 2)*2^(2n-k);
a(n) = Sum_{k=0..n} (binomial(2n-k, k) mod 2)*2^k. (End)
a(n) = Sum_{k=0..2n} A106344(2n,k)*2^(2n-k). - Philippe Deléham, Dec 18 2008

A099093 Riordan array (1, 3+3x).

Original entry on oeis.org

1, 0, 3, 0, 3, 9, 0, 0, 18, 27, 0, 0, 9, 81, 81, 0, 0, 0, 81, 324, 243, 0, 0, 0, 27, 486, 1215, 729, 0, 0, 0, 0, 324, 2430, 4374, 2187, 0, 0, 0, 0, 81, 2430, 10935, 15309, 6561, 0, 0, 0, 0, 0, 1215, 14580, 45927, 52488, 19683, 0, 0, 0, 0, 0, 243, 10935, 76545, 183708, 177147, 59049
Offset: 0

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Author

Paul Barry, Sep 25 2004

Keywords

Comments

Row sums are A030195. Diagonal sums are A099094.
The Riordan array (1,s+tx) defines T(n,k) = binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

Examples

			Rows begin:
  1;
  0, 3;
  0, 3, 9;
  0, 0, 18, 27;
  0, 0, 9, 81, 81;
  0, 0, 0, 81, 324, 243;
  0, 0, 0, 27, 486, 1215, 729;
  ...

Crossrefs

Cf. A038221.

Programs

  • Magma
    [[Binomial(k,n-k)*3^k: k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Feb 21 2015 /* as the triangle */
  • PARI
    tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(k, n-k)*3^k, ", ");); print(););} \\ Michel Marcus, Feb 21 2015

Formula

T(n,k) = binomial(k, n-k)*3^k. - corrected by Michel Marcus, Feb 21 2015
Columns have g.f. (3x+3x^3)^k.
T(n,k) = A026729(n,k)*3^k. - Philippe Deléham, Jul 29 2006

A130167 Another version of triangle in A127743.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 22, 16, 9, 4, 1, 0, 92, 60, 31, 14, 5, 1, 0, 426, 252, 120, 52, 20, 6, 1, 0, 2146, 1160, 510, 209, 80, 27, 7, 1, 0, 11624, 5776, 2348, 904, 335, 116, 35, 8, 1, 0, 67146, 30832, 11610, 4184, 1481, 507, 161, 44, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Aug 03 2007

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,1,2,1,3,1,4,1,5,1,6,1,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
A154380*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 13 2009

Examples

			Triangle begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  6,  5,  3,  1;
  0, 22, 16,  9,  4,  1;
  0, 92, 60, 31, 14,  5,  1; ...

Crossrefs

Cf. A074664.

Formula

Sum_{k=0..n} T(n,k) = A000110(n).

A112883 A skew Jacobsthal-Pascal matrix.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 2, 5, 0, 0, 1, 7, 11, 0, 0, 0, 3, 16, 21, 0, 0, 0, 1, 12, 41, 43, 0, 0, 0, 0, 4, 34, 94, 85, 0, 0, 0, 0, 1, 18, 99, 219, 171, 0, 0, 0, 0, 0, 5, 60, 261, 492, 341, 0, 0, 0, 0, 0, 1, 25, 195, 678, 1101, 683, 0, 0, 0, 0, 0, 0, 6, 95, 576, 1692, 2426, 1365, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul Barry, Oct 05 2005

Keywords

Comments

T(n,n) is A001045(n), row sums are A006130, column sums are A002605. Compare with [0,1,-1,0,0,..] DELTA [1,2,-2,0,0,...] where DELTA is the operator defined in A084938. A skewed version of the Riordan array (1/(1-x-2x^2),x/(1-x-2x^2)) (A073370).
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

Examples

			Rows begin
  1;
  0, 1;
  0, 1, 3;
  0, 0, 2, 5;
  0, 0, 1, 7, 11;
  0, 0, 0, 3, 16, 21;
  0, 0, 0, 1, 12, 41, 43;
  0, 0, 0, 0,  4, 34, 94,  85;
  0, 0, 0, 0,  1, 18, 99, 219, 171;
  0, 0, 0, 0,  0,  5, 60, 261, 492,  341;
  0, 0, 0, 0,  0,  1, 25, 195, 678, 1101, 683;

Crossrefs

Cf. A111006.

Formula

From Philippe Deléham: (Start)
G.f.: 1/(1-yx(1-x)-2x^2*y*2);
Number triangle T(n, k) = Sum_{j=0..2k-n} C(n-k+j, n-k)*C(j, 2k-n-j)*2^(2k-n-j);
T(n, k) = A073370(k, n-k); T(n, k) = T(n-1, k-1) + T(n-2, k-1) + 2*T(n-2, k-2). (End)
Previous Showing 11-14 of 14 results.

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