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.

A339371 Number of partitions of n into an even number of Fibonacci parts (with a single type of 1).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 4, 8, 7, 13, 12, 18, 18, 27, 27, 39, 38, 53, 53, 72, 73, 96, 98, 126, 128, 165, 168, 209, 216, 266, 274, 334, 345, 416, 430, 514, 533, 628, 655, 766, 797, 929, 966, 1115, 1164, 1336, 1395, 1590, 1661, 1885, 1969, 2226, 2326, 2611, 2734
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(7) = 4 because we have [5, 2], [3, 2, 1, 1], [2, 2, 2, 1] and [2, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000045, A003107, A027187, A093997, A093998, A298949, A339372.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[(1/2) (Product[1/(1 - x^Fibonacci[k]), {k, 2, 26}] + Product[1/(1 + x^Fibonacci[k]), {k, 2, 26}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=2} 1 / (1 - x^Fibonacci(k)) + Product_{k>=2} 1 / (1 + x^Fibonacci(k))).
a(n) = (A003107(n) + A298949(n)) / 2.

A339372 Number of partitions of n into an odd number of Fibonacci parts (with a single type of 1).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 6, 6, 10, 9, 15, 15, 23, 22, 32, 32, 45, 46, 62, 62, 84, 84, 110, 113, 144, 147, 185, 191, 237, 243, 299, 308, 372, 387, 462, 479, 569, 591, 695, 723, 843, 879, 1017, 1063, 1222, 1273, 1459, 1523, 1732, 1812, 2048, 2141, 2411, 2523, 2830
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(8) = 6 because we have [8], [5, 2, 1], [3, 3, 2], [3, 2, 1, 1, 1], [2, 2, 2, 1, 1] and [2, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000045, A003107, A027193, A093997, A093998, A298949, A339371.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[(1/2) (Product[1/(1 - x^Fibonacci[k]), {k, 2, 26}] - Product[1/(1 + x^Fibonacci[k]), {k, 2, 26}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=2} 1 / (1 - x^Fibonacci(k)) - Product_{k>=2} 1 / (1 + x^Fibonacci(k))).
a(n) = (A003107(n) - A298949(n)) / 2.

A357381 Expansion of Product_{k>=1} 1 / (1 + x^Fibonacci(k)).

Original entry on oeis.org

1, -2, 2, -3, 5, -7, 9, -11, 13, -16, 20, -23, 26, -31, 36, -41, 48, -55, 62, -71, 81, -92, 104, -116, 129, -145, 163, -180, 198, -219, 242, -267, 293, -320, 349, -381, 416, -452, 489, -529, 572, -618, 668, -719, 771, -829, 892, -956, 1023, -1094, 1167, -1246, 1331, -1416, 1504
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2022

Keywords

Comments

Convolution inverse of A000121.

Crossrefs

Cf. A000045, A000121, A298949, A357380.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Product[1/(1 + x^Fibonacci[k]), {k, 1, 21}], {x, 0, nmax}], x]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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