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 1-3 of 3 results.

A158945 Triangle read by rows, A158944 * an infinite matrix with A158943 (prefaced with a 1) as the right border: (1, 1, 1, 3, 5, 10, 19, 36, ...) and the rest zeros.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 3, 0, 2, 0, 5, 0, 3, 0, 6, 0, 10, 4, 0, 3, 0, 10, 0, 19, 0, 4, 0, 9, 0, 20, 0, 36, 5, 0, 4, 0, 15, 0, 38, 0, 69, 0, 5, 0, 12, 0, 30, 0, 72, 0, 131, 6, 0, 5, 0, 20, 0, 57, 0, 138, 0, 250, 0, 6, 0, 15, 0, 40, 0, 108, 0, 262, 0, 476
Offset: 1

Views

Author

Gary W. Adamson, Mar 31 2009

Keywords

Comments

As a property of eigentriangles, sum of n-th row terms = rightmost term of next row. Right border = A158943 prefaced with a 1: (1, 1, 1, 3, 5, 10, 19, 36, 69, ...).

Examples

			First few rows of the triangle:
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0,  3;
  3, 0, 2,  0,  5;
  0, 3, 0,  6,  0, 10;
  4, 0, 3,  0, 10,  0, 19;
  0, 4, 0,  9,  0, 20,  0,  36;
  5, 0, 4,  0, 15,  0, 38,   0,  69;
  0, 5, 0, 12,  0, 30,  0,  72,   0, 131;
  6, 0, 5,  0, 20,  0, 57,   0, 138,   0, 250;
  0, 6, 0, 15,  0, 40,  0, 108,   0, 262,   0, 476;
  7, 0, 6,  0, 25,  0, 76,   0, 207,   0, 500,   0, 907;
  ...
Row 5 = (3, 0, 2, 0, 5) = termwise products of (3, 0, 2, 0, 1) and (1, 1, 1, 3, 5); where (3, 0, 2, 0, 1) = row 5 of triangle A158944.

Crossrefs

Cf. A158943, A158944.

Formula

Triangle read by rows, A158944 * an infinite matrix with A158943 (prefaced with a 1) as the right border: (1, 1, 1, 3, 5, 10, 19, 36, ...) and the rest zeros.

A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

Original entry on oeis.org

1, 1, 3, 5, 10, 19, 36, 69, 131, 250, 476, 907, 1728, 3292, 6272, 11949, 22765, 43371, 82629, 157422, 299915, 571388, 1088589, 2073943, 3951206, 7527704, 14341527, 27322992, 52054840, 99173120, 188941273, 359964521, 685792227, 1306548149
Offset: 1

Views

Author

Gary W. Adamson, Mar 31 2009

Keywords

Comments

Equals row sums of triangle A158945.
Number of compositions of n into odd parts where there is 1 sort of part 1, 2 sorts of part 3, 3 sorts of part 5, ..., k sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

Examples

			The INVERT transform of (1, N, ...) begins (1, (N+1), ...) so that we have (1, 1, ...) placed in ascending magnitude in the bottom row. In the top row we place an equal number of descending terms: (..., 0, 3, 0, 2, 0, 1). Take the dot product of terms in top and bottom rows, adding the result to the next term A027656: (1, 0, 2, 0, 3, ...). a(6) = 19 given: 3, 0, 2, 0, 1 1, 1, 3, 5, 10 Dot product of top row terms * bottom row terms = (1, 0, 2, 0, 3) dot (1, 1, 3, 5, 10) = (3 + 0 + 6 + 0 + 10) = 19, which is added to the next term in (1, 0, 2, 0, 3, ...); i.e., (an 0) = 19.

Links

Crossrefs

Cf. A027656, A052535, A158944, A158945.

Programs

  • GAP
    a:=[1,1,3,5];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jul 12 2019
  • Magma
    I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 09 2019
  • Maple
    A027656 := proc(n) if type(n,odd) then 0; else n/2+1 ; fi; end: L := [seq(A027656(n), n=0..100)] ; read("transforms"); INVERT(L) ; # R. J. Mathar, Apr 02 2009
  • Mathematica
    LinearRecurrence[{1, 2, 0, -1}, {1, 1, 3, 5}, 40] (* Vincenzo Librandi, Jul 09 2019 *)
  • PARI
    my(x='x+O('x^40)); Vec(x/(1-x-2*x^2+x^4)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    a=(x/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 12 2019

Formula

INVERT transform of (1, 0, 2, 0, 3, 0, 4, ...); i.e., the natural numbers interleaved with zeros.
From R. J. Mathar, Apr 02 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-4).
G.f.: x/(1 - x - 2*x^2 + x^4). (End)
The sequence is the second INVERT transform of (1, -1, 3, -5, 10, -19, ...). - Gary W. Adamson, Jul 08 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k). - Seiichi Manyama, Jun 12 2024

Extensions

Extended by R. J. Mathar, Apr 02 2009

A156663 Triangle by columns, powers of 2 interleaved with zeros.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 4, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 8, 0, 4, 0, 2, 0, 1, 0, 8, 0, 4, 0, 2, 0, 1, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1
Offset: 0

Views

Author

Gary W. Adamson, Feb 12 2009

Keywords

Comments

Eigensequence of the triangle = A001045.

Examples

			First few rows of the triangle =
   1;
   0,  1;
   2,  0,  1;
   0,  2,  0,  1;
   4,  0,  2,  0,  1;
   0,  4,  0,  2,  0,  1;
   8,  0,  4,  0,  2,  0,  1;
   0,  8,  0,  4,  0,  2,  0,  1;
  16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   0,  0, -2,  0,  1;
   0,  0,  0, -2,  0,  1;
   0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0,  0, -2,  0,  1;
   ... - _Peter Bala_, Aug 15 2021

Links

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

Crossrefs

Cf. A001045, A158944.

Programs

  • Maple
    seq(seq( sqrt(2)^(n-k) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Formula

Triangle by columns, (1, 0, 2, 0, 4, 0, 8, ...) in every column.
From Peter Bala, Aug 15 2021: (Start)
T(n,k) = sqrt(2)^((n - k)/2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
Double Riordan array (1/(1 - 2*x^2); x, x) as defined in Davenport et al.
The m-th power of the array is the double Riordan array (1/(1 - 2*x^2)^(m); x, x). Cf. A158944. (End)

Extensions

Typo in Data corrected by Peter Bala, Aug 15 2021
Showing 1-3 of 3 results.

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