A037667 -id:A037667 - cp's OEIS frontend

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.

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

Jonathan Shadrach Gilbert, Nov 28 2024

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

Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).

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

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

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

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

Jonathan Shadrach Gilbert, Nov 28 2024

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

Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).

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

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

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

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

Jonathan Shadrach Gilbert, Dec 03 2024

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

Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).

Cf. A007090, A378497, A378499, A037667, A037618.

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

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.

cp's OEIS Frontend

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.

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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.

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

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.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula