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.

User: Jonathan Shadrach Gilbert

Jonathan Shadrach Gilbert's wiki page.

Jonathan Shadrach Gilbert has authored 3 sequences.

A378502 a(n) is the number whose base-4 digits are 1 followed by the first n-1 terms of the periodic sequence with initial period 3,2,0.

Original entry on oeis.org

1, 7, 30, 120, 483, 1934, 7736, 30947, 123790, 495160, 1980643, 7922574, 31690296, 126761187, 507044750, 2028179000, 8112716003, 32450864014, 129803456056, 519213824227, 2076855296910, 8307421187640, 33229684750563, 132918739002254, 531674956009016
Offset: 1

Author

Keywords

Examples

			a(7) = 1320320_4 = 7736.
a(8) = 13203203_4 = 30947.
		

Crossrefs

Programs

  • Mathematica
    Table[FromDigits[Join[{1}, PadRight[{}, n-1, {3, 2, 0}]], 4], {n, 30}]

Formula

a(n) = floor((17/36)*4^n).
G.f.: -x*(x^3-2*x^2-3*x-1)/((x-1)*(4*x-1)*(x^2+x+1)).
a(n) = 4^(n-1) + A037667(n-1) for n > 1.

A378499 a(n) is the number whose base-4 digits are 1 followed by the first n-1 terms of the periodic sequence with initial period 0,3,2.

Original entry on oeis.org

1, 4, 19, 78, 312, 1251, 5006, 20024, 80099, 320398, 1281592, 5126371, 20505486, 82021944, 328087779, 1312351118, 5249404472, 20997617891, 83990471566, 335961886264, 1343847545059, 5375390180238, 21501560720952, 86006242883811, 344024971535246
Offset: 1

Author

Keywords

Examples

			a(7) = 1032032_4 = 5006.
a(8) = 10320320_4 = 20024.
		

Crossrefs

Cf.: A378497, A037667, A037618, A037681, A378502. Periodic part of sequence rotated left relative to A378497.

Programs

  • Mathematica
    Table[FromDigits[Join[{1}, PadRight[{}, n-1, {0, 3, 2}]], 4], {n, 30}]

Formula

a(n) = floor((11/36)*4^n).
G.f.: z*(z^3+3*z^2+1)/((z-1)*(4*z-1)*(z^2+z+1)).
E.g.f.: (9 - 20*exp(x) + 11*exp(4*x) + 8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/36. - Stefano Spezia, Nov 29 2024

A378497 a(n) is the number whose base-4 digits are 1 followed by the first n-1 terms of the periodic sequence with initial period 2,0,3.

Original entry on oeis.org

1, 6, 24, 99, 398, 1592, 6371, 25486, 101944, 407779, 1631118, 6524472, 26097891, 104391566, 417566264, 1670265059, 6681060238, 26724240952, 106896963811, 427587855246, 1710351420984, 6841405683939, 27365622735758, 109462490943032, 437849963772131
Offset: 1

Author

Keywords

Examples

			a(7) = 1203203_4 = 6371.
a(8) = 12032032_4 = 25486.
		

Crossrefs

Cf. A378499, A037667, A037618, A378502. Periodic part of sequence rotated right relative to A378499. Same first four terms as A037688, and similar formula.

Programs

  • Mathematica
    Table[FromDigits[Join[{1},PadRight[{}, n-1, {2, 0, 3}]], 4], {n, 30}]

Formula

a(n) = floor((7/18)*4^n).
G.f.: z*(2*z^3+2*z+1)/((z-1)*(4*z-1)*(z^2+z+1)).
E.g.f.: (9 - 10*exp(x) + 7*exp(4*x) - 2*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Nov 29 2024