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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A295724 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 2.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 18, 31, 57, 96, 169, 281, 482, 795, 1341, 2200, 3669, 5997, 9922, 16175, 26609, 43296, 70929, 115249, 188226, 305523, 497845, 807464, 1313501, 2129157, 3459042, 5604583, 9096393, 14733744, 23895673, 38694953, 62721698, 101547723, 164531565
Offset: 0

Views

Author

Clark Kimberling, Nov 29 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {0, 0, 1, 2}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 2.
G.f.: (x^2 (1 + x))/((-1 + x + x^2) (-1 + 2 x^2)).

A295725 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = 0, a(2) = -1, a(3) = 1.

Original entry on oeis.org

0, 0, -1, 1, -2, 3, -3, 8, -3, 21, 2, 55, 25, 144, 105, 377, 354, 987, 1085, 2584, 3157, 6765, 8898, 17711, 24561, 46368, 66833, 121393, 180034, 317811, 481461, 832040, 1280733, 2178309, 3393506, 5702887, 8965321, 14930352, 23633529, 39088169, 62197410
Offset: 0

Views

Author

Clark Kimberling, Nov 29 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {0, 0, -1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 0, a(1) = 0, a(2) = -1, a(3) = 1.
G.f.: (-1 + x)/(-1 + x + x^2) + 1/(-1 + 2 x^2).

A295726 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = -1, a(2) = 1, a(3) = 1.

Original entry on oeis.org

0, -1, 1, 1, 6, 9, 23, 36, 75, 119, 226, 361, 651, 1044, 1823, 2931, 5010, 8069, 13591, 21916, 36531, 58959, 97538, 157521, 259155, 418724, 686071, 1108891, 1811346, 2928429, 4772543, 7717356, 12555435, 20305559, 32992066, 53363161, 86617371, 140111604
Offset: 0

Views

Author

Clark Kimberling, Nov 29 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {0, -1, 1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 0, a(1) = -1, a(2) = 1, a(3) = 1.
G.f.: (-x + 2 x^2 + 3 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295727 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -1, a(1) = 1, a(2) = 1, a(3) = 1.

Original entry on oeis.org

-1, 1, 1, 1, 4, 3, 11, 10, 29, 31, 76, 91, 199, 258, 521, 715, 1364, 1951, 3571, 5266, 9349, 14103, 24476, 37555, 64079, 99586, 167761, 263251, 439204, 694263, 1149851, 1827730, 3010349, 4805311, 7881196, 12620971, 20633239, 33123138, 54018521, 86879515
Offset: 0

Views

Author

Clark Kimberling, Nov 29 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {-1, 1, 1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = -1, a(1) = 1, a(2) = 1, a(3) = 1.
G.f.: (1 - 3 x)/(-1 + x + x^2) + x/(-1 + 2 x^2).

A295728 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 1, a(1) = -1, a(2) = 1, a(3) = 1.

Original entry on oeis.org

1, -1, 1, 1, 4, 7, 15, 26, 49, 83, 148, 247, 427, 706, 1197, 1967, 3292, 5387, 8935, 14578, 24025, 39115, 64164, 104303, 170515, 276866, 451477, 732439, 1192108, 1932739, 3141231, 5090354, 8264353, 13387475, 21717364, 35170375, 57018811, 92320258, 149601213
Offset: 0

Views

Author

Clark Kimberling, Nov 29 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {1, -1, 1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = -1, a(2) = 1, a(3) = 1.
G.f.: (1 - 2 x - x^2 + 5 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295729 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -1, a(1) = 0, a(2) = 1, a(3) = 1.

Original entry on oeis.org

-1, 0, 1, 1, 6, 7, 21, 28, 65, 93, 190, 283, 537, 820, 1485, 2305, 4046, 6351, 10909, 17260, 29193, 46453, 77694, 124147, 205937, 330084, 544213, 874297, 1434894, 2309191, 3776853, 6086044, 9928433, 16014477, 26073982, 42088459, 68424585, 110513044
Offset: 0

Views

Author

Clark Kimberling, Nov 30 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {-1, 0, 1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = -1, a(1) = 0, a(2) = 1, a(3) = 1.
G.f.: (-1 + x + 4 x^2 - 2 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295730 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -1, a(1) = 0, a(2) = 0, a(3) = 1.

Original entry on oeis.org

-1, 0, 0, 1, 3, 6, 13, 23, 44, 75, 135, 226, 393, 651, 1108, 1823, 3059, 5010, 8325, 13591, 22428, 36531, 59983, 97538, 159569, 259155, 422820, 686071, 1117083, 1811346, 2944813, 4772543, 7750124, 12555435, 20371095, 32992066, 53494233, 86617371, 140373748
Offset: 0

Views

Author

Clark Kimberling, Nov 30 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {-1, 0, 0, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = -1, a(1) = 0, a(2) = 0, a(3) = 1.
G.f.: (-1 + x + 3 x^2 - x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295731 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -1, a(1) = -1, a(2) = 0, a(3) = 1.

Original entry on oeis.org

-1, -1, 0, 1, 5, 10, 23, 41, 80, 137, 249, 418, 731, 1213, 2072, 3413, 5741, 9410, 15663, 25585, 42272, 68881, 113201, 184130, 301427, 489653, 799272, 1297117, 2112773, 3426274, 5571815, 9030857, 14668208, 23764601, 38563881, 62459554, 101285579, 164007277
Offset: 0

Views

Author

Clark Kimberling, Nov 30 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {-1, -1, 0, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = -1, a(1) = -1, a(2) = 0, a(3) = 1.
G.f.: (-1 + 4 x^2 + 2 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295732 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -1, a(1) = -1, a(2) = -1, a(3) = 1.

Original entry on oeis.org

-1, -1, -1, 1, 2, 9, 15, 36, 59, 119, 194, 361, 587, 1044, 1695, 2931, 4754, 8069, 13079, 21916, 35507, 58959, 95490, 157521, 255059, 418724, 677879, 1108891, 1794962, 2928429, 4739775, 7717356, 12489899, 20305559, 32860994, 53363161, 86355227, 140111604
Offset: 0

Views

Author

Clark Kimberling, Nov 30 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {-1, -1, -1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = -1, a(1) = -1, a(2) = -1, a(3) = 1.
G.f.: (-1 + 3 x^2 + 3 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

A295733 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = -1, a(2) = -1, a(3) = 1.

Original entry on oeis.org

0, -1, -1, 1, 0, 7, 7, 26, 33, 83, 116, 247, 363, 706, 1069, 1967, 3036, 5387, 8423, 14578, 23001, 39115, 62116, 104303, 166419, 276866, 443285, 732439, 1175724, 1932739, 3108463, 5090354, 8198817, 13387475, 21586292, 35170375, 56756667, 92320258, 149076925
Offset: 0

Views

Author

Clark Kimberling, Nov 30 2017

Keywords

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Crossrefs

Cf. A001622, A000045, A005672.

Programs

  • Mathematica
    LinearRecurrence[{1, 3, -2, -2}, {0, -1, -1, 1}, 100]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 0; a(1) = -1, a(2) = -1, a(3) = 1.
G.f.: (-x + 5 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).
Previous Showing 11-20 of 23 results. Next

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