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.

Previous Showing 51-55 of 55 results.

A250004 Numbers representable as x^y + x*y, where x>1, y>1 are integers (without multiplicity).

Original entry on oeis.org

8, 14, 15, 24, 35, 36, 42, 48, 63, 76, 80, 93, 99, 120, 140, 142, 143, 168, 195, 224, 234, 255, 258, 272, 288, 323, 360, 364, 399, 440, 483, 528, 530, 536, 575, 624, 645, 675, 728, 747, 756, 783, 840, 899, 960, 1023, 1030, 1044, 1088, 1155, 1224, 1295, 1320, 1364, 1368
Offset: 1

Views

Author

Alex Ratushnyak, Jan 14 2015

Keywords

Comments

The sequence of numbers representable as x^y + x*y in two or more ways begins: 24, 76, 272, 1044, 2208, 4120, 16412. Example: 2208 = 46^2 + 46*2 = 3^7 + 3*7.
The subsequence of squares begins: 36, 1764.

Examples

			a(2) = 14 = 2^3 + 2*3.
a(3) = 15 = 3^2 + 3*2.
a(22) = 255 = 15^2 + 15*2.

Crossrefs

Cf. A253775.
Cf. A024036 is a subsequence, except first 2 terms.

Programs

  • PARI
    isok(n) = {for (p=2, floor(log(n)/log(2)), for (k=2, sqrtnint(n, p), if (n == k^p + p*k, return (1)););); return (0);} \\ Michel Marcus, Jan 16 2015

A334921 Expansion of Phi(x) = (1/(1+x))*Product_{k>=0} (1-(x/(1+x))^2^k).

Original entry on oeis.org

1, -2, 2, 0, -6, 20, -48, 96, -166, 252, -340, 416, -480, 544, -544, 0, 2906, -13396, 44100, -121792, 296860, -652808, 1306560, -2377280, 3879136, -5461952, 5892512, -2171520, -11699616, 45871040, -114213888, 228427776, -377994406, 478195212, -252252460, -1013309824
Offset: 0

Views

Author

Michel Marcus, May 16 2020

Keywords

Comments

The Hankel transforms of Phi(x) and Phi(x^2) are identical. See Theorem 2.8 in Han paper.

Links

Crossrefs

Cf. A000120 (Hamming weight of n), A024036 (4^n - 1).

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * (-1)^(2*n - 2*k + DigitCount[k, 2, 1]), {k, 0, n}]; Array[a, 36, 0] (* Amiram Eldar, May 16 2020 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+hammingweight(k)));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+A000120(k)). See Theorem 2.8 in Han paper.
a(n) = 0 for positive terms of A024036.

A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward.

Original entry on oeis.org

1, 0, 3, 1, 7, 3, 15, 7, 31, 15, 63, 31, 127, 63, 255, 127, 511, 255, 1023, 511, 2047, 1023, 4095, 2047, 8191, 4095, 16383, 8191, 32767, 16383, 65535, 32767, 131071, 65535, 262143, 131071, 524287, 262143, 1048575, 524287, 2097151, 1048575, 4194303, 2097151, 8388607, 4194303
Offset: 0

Views

Author

Jim Singh, Jul 12 2018

Keywords

Examples

			Let 1. The first four terms are 1, (1-1)/2 = 0, 2*1+1 = 3, 1.
Let 4*1+3 = 7. The next four terms are 7, (7-1)/2 = 3, 2*7+1 = 15, 7.
Let 4*7+3 = 31. The next four terms are 31, (31-1)/2 = 15, 2*31+1 = 63, 31; etc.

Links

Crossrefs

Cf. A000225, A024036, A083420.

Programs

  • GAP
    a:=[1,0,3];; for n in [4..45] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # Muniru A Asiru, Jul 14 2018
  • Maple
    seq(coeff(series((1-x+x^2)/((1-x)*(1-2*x^2)), x,n+1),x,n),n=0..45); # Muniru A Asiru, Jul 14 2018
  • Mathematica
    CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - 2 x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Jul 13 2018 *)
LinearRecurrence[{1, 2, -2}, {1, 0, 3}, 46] (* Robert G. Wilson v, Jul 21 2018 *)

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n>2, a(0)=1, a(1)=0, a(2)=3.
From Bruno Berselli, Jul 12 2018: (Start)
G.f.: (1 - x + x^2)/((1 - x)*(1 - 2*x^2)).
a(n) = 2*a(n-2) + 1 for n>1, a(0)=1, a(1)=0.
a(n) = (1 + (-1)^n)*(2^(n/2) - 2^((n-3)/2)) + 2^((n-1)/2) - 1.
Therefore: a(4*k) = 2*4^k - 1, a(4*k+1) = 4^k - 1, a(4*k+2) = 4^(k+1) - 1, a(4*k+3) = 2*4^k - 1. (End)

A333813 a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

Original entry on oeis.org

0, 0, 6, 4, 46, 12, 294, 1908, 1630, 13084, 6486, 84996, 517134, 502828, 3605638, 2428308, 24062142, 5077564, 149450422, 985222180, 808182894, 6719515980, 2978678758, 43295774644, 267326277406, 252223018332, 1856180682774, 1170495537220
Offset: 0

Views

Author

Ctibor O. Zizka, Apr 06 2020

Keywords

Comments

For integers X, Y, let a(n) = (X^(t+1) - 1) / (X - 1) - Y^n, where t = floor(n*log_X(Y)) . This sequence is for X = 2, Y = 3.

Examples

			a(0) = 2^(1 + floor(0*log_2(3))) - (3^0 + 1) = 0; a(4) = 2^(1 + floor(4*log_2(3))) - (3^4 + 1) = 46.

Crossrefs

Cf. A000225, A024036.
Examples for integers X = Y from {2, 3, 4, 5, 6, 7, 8, 9, 10} are A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275. Examples for X = 2, Y = 4 are A024036; for X = 2, Y = 8, A024088; and for X = 3, Y = 9, A191681.

Programs

  • Mathematica
    Table[2^(1+Floor[n Log2[3]])-(3^n+1),{n,0,30}] (* Harvey P. Dale, Sep 04 2023 *)

Formula

a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

A346142 a(n) is the number of linear orthogonal cellular automata pairs with maximal period 2^2n-1.

Original entry on oeis.org

1, 1, 3, 15, 42, 181, 572, 1872, 5899
Offset: 3

Views

Author

Michael De Vlieger, Oct 01 2021

Keywords

Links

Crossrefs

Cf. A002450, A024036.
Previous Showing 51-55 of 55 results.

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