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 1-5 of 5 results.

A160646 First differences of A160644.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, 77, 104, 148, 197, 265, 363, 481, 638, 858, 1126, 1480, 1953, 2544, 3309, 4312, 5566, 7175, 9246, 11843, 15136, 19328, 24564, 31158, 39466, 49811, 62737, 78900, 98931, 123817, 154707, 192830, 239911, 298013
Offset: 1

Views

Author

Alford Arnold, May 24 2009

Keywords

Comments

Second differences count a subset of unrestricted partitions; cf. A160648.

Examples

			A160644 begins 1, 1, 2, 3, 4, 7, 10, 14, 22, 32, 45, 67, 95, 134, 192, ... so a(n) begins 0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, ...

Links

Crossrefs

Cf. A000041, A002685, A053445, A160644, A160648.

A160648 Second differences of sequence A160644.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 4, 2, 3, 9, 6, 11, 19, 19, 27, 44, 49, 68, 98, 118, 157, 220, 268, 354, 473, 591, 765, 1003, 1254, 1609, 2071, 2597, 3293, 4192, 5236, 6594, 8308, 10345, 12926, 16163, 20031, 24886, 30890, 38123, 47081, 58102, 71381, 87704, 107643
Offset: 1

Views

Author

Alford Arnold, May 25 2009

Keywords

Comments

A160644 bisects sequence A053445 which counts unrestricted partitions such that the two largest values match and that no part is less than two.
Conjecture: a(n) counts unrestricted partitions of even numbers such that
the three largest values match and that, after "222", no part is less than three.

Examples

			a(n) begins 0 1 0 0 2 0 1 4 2 3 9 ... and counts 222; 444,3333;
666,5553,444433,333333; 5555,44444; 6664,55543,444433;
888,6666,7773,55554,66633,444444,555333,4443333,33333333; ...

Links

Crossrefs

Cf. A000041, A002865, A053445, A160644, A160646.

A161922 Table with the mapped A125106(p) in row n where p runs through the partitions counted by A160644(n).

Original entry on oeis.org

2, 6, 12, 14, 24, 26, 30, 48, 50, 54, 62, 56, 60, 96, 98, 102, 110, 126, 104, 108, 114, 122, 192, 194, 198, 206, 222, 254, 120, 200, 204, 210, 218, 230, 246, 384, 386, 390, 398, 414, 446, 510, 216, 224, 228, 236, 242, 252, 392, 396, 402, 410, 422, 438, 462, 494, 768, 770
Offset: 1

Views

Author

Alford Arnold, Jul 06 2009

Keywords

Comments

A160644(n) with n > 0 counts the partitions of 2n such that all parts are > 1 and the largest part occurs more than once. If n=7, these are 10 partitions of 14: 2^7 = (2^4;3^2) = (2^1;3^4) = (2^3;4^2) = (3^2;4^2) = (2^1;4^3) = (2^2;5^2) = (4^1;5^2) = (2^1;6^2) = 7^2, for example.
For each of these admitted partitions p of 2n, p is mapped to a binary and the decimal rep. of this binary is added to row n of this table here, sorting the row according to the natural order of integers (not according to any property of partitions).

Examples

			The partition 4+4+4+4 = 16 and maps to 120 = 64 + 32 + 16 + 8 as described in A125106, so 120 is in the 8th row.
The table has A160644(n) integers in row n and starts
2,
6,.......[2,2]->6
12,14,..........[3,3]->12, [2,2,2]->14
24,26,30,...........[4,4]->24, [2,3,3]->26, [2,2,2,2] ->30
48,50,54,62, ....... [5,5]->48, [2,4,4]->50, [2,2,3,3]->54, [2,2,2,2,2]->62
56,60,96,98,102,110,126,.....[4,4,4]->56, [3,3,3,3]->60, [6,6]->96, [2,5,5]->98, [2,2,4,4]->102, [2,2,2,3,3]->110
104,108,114,122,192,194,198,206,222,254,...[4,5,5]->104, [3,3,4,4]->108, [2,4,4,4]->114, [2,3,3,3,3]->122

Programs

  • Maple
    A125106m := proc(par) local c,dgs,p ; c := 1 ; dgs := [] ; for p in par do if p = c then dgs := [op(dgs),1] ; else dgs := [op(dgs),seq(0,j=1..p-c),1] ; fi; c := p ; od: add(op(i,dgs) *2^(i-1), i=1..nops(dgs)) ; end:
A161922 := proc(n) r := {} ; prts := combinat[partition](2*n) ; for p in prts do convert(p,set) intersect {1}; if % = {} then if nops(p) < 2 then ; elif op(-1,p) = op(-2,p) then r := r union {A125106m(p)} ; fi; fi; od: sort(r) ; end:
for n from 1 to 11 do A161922(n) ; od; # R. J. Mathar, Sep 11 2009

Extensions

Detailed description and examples and rows n >= 8 completed by R. J. Mathar, Sep 11 2009

A160643 Bisect A053445 then calculate the first differences of the resulting sequence.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 6, 11, 15, 20, 33, 43, 60, 88, 119, 160, 226, 300, 404, 549, 727, 961, 1283, 1680, 2201, 2887, 3750, 4857, 6301, 8105, 10410, 13357, 17050, 21714, 27625, 34992, 44240, 55840, 70261, 88220, 110600, 138274, 172558, 214984, 267234
Offset: 1

Views

Author

Alford Arnold, May 25 2009, Jun 20 2009

Keywords

Comments

a(n) counts the following subset of the partitions (cf. A000041): the number being partitioned is odd, the minimum part is two
and the three largest parts match.
First differences of A161921.

Examples

			A161921 begins: 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, ...
Therefore a(n) begins 0, 0, 0, 1, 1, 1, 4, 4, 6, ..., counting 333; 3332; 33322; 555, 4443, 333222, 33333; etc.

Links

Crossrefs

Cf. A053445, A160644.

Programs

  • Mathematica
    Join[{0},Differences[Take[Differences[Table[PartitionsP[n],{n,0,100}],2],{2,-1,2}]]] (* Harvey P. Dale, Sep 02 2013 *)

Extensions

Extended and edited by Nathaniel Johnston, Apr 30 2011

A161921 The bisection A053445(2n+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, 792, 1092, 1496, 2045, 2772, 3733, 5016, 6696, 8897, 11784, 15534, 20391, 26692, 34797, 45207, 58564, 75614, 97328, 124953, 159945, 204185, 260025, 330286, 418506, 529106, 667380, 839938
Offset: 0

Views

Author

Alford Arnold, Jul 05 2009

Keywords

Links

Crossrefs

Cf. A160644 (the other bisection), A160643 (first differences of a(n)).

Programs

  • Mathematica
    Take[Differences[Table[PartitionsP[n],{n,0,100}],2],{2,-1,2}] (* Harvey P. Dale, Sep 02 2013 *)
Showing 1-5 of 5 results.

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