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.

A064544 Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 143, 145, 154, 156, 165, 167, 176, 178, 187, 189, 198, 202, 211, 213, 220, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 303, 312, 314, 321, 325, 330, 336, 341, 347, 352, 358
Offset: 0

Views

Author

David W. Wilson, Oct 09 2001

Keywords

Comments

This sequence is 10-automatic (decimal expansions form a regular language accepted by a finite automaton).

Examples

			143 is in the sequence because its digits {1, 4, 3} may be grouped so that 1+3 = 4.

Links

Crossrefs

Cf. A064671, A064686 (number of n-digit base-3 biquams), A065023, A065024, A065025.

Programs

  • PARI
    is(n) = { my (d=digits(n), s=[0]); for (k=1, #d, s=setunion(apply(v -> v+d[k], s), apply(v -> v-d[k], s))); setsearch(s, 0)>0 } \\ Rémy Sigrist, Jan 23 2021

A065024 Number of n-digit biquanimous numbers in base 10 allowing leading zeros.

Original entry on oeis.org

1, 10, 136, 2056, 29246, 376414, 4366881, 47111408, 487875964, 4951921240, 49815780829, 499304300676, 4997363405880, 49989815235610, 499959437775564, 4999832460244272, 49999282163551040, 499996822399017380, 4999985554326500949, 49999932964605448756, 499999684083134646700, 4999998493912339729030, 49999992756990963293576, 499999964931001199898296, 4999999829289953917354596
Offset: 1

Views

Author

N. J. A. Sloane, Nov 03 2001

Keywords

Comments

A biquanimous number (A064544) is a number whose digits can be split into two groups with equal sums.

References

Links

  • Alois P. Heinz, Table of n, a(n) for n = 1..1000
  • Index entries for linear recurrences with constant coefficients, signature (62, -1807, 33062, -427564, 4169600, -31932484, 197416064, -1004816182, 4272066348, -15337434186, 46879240956, -122734147260, 276448013616, -537280650948, 902485024560, -1310712845937, 1644560278758, -1778909274239, 1653055768558, -1312795678832, 884596325632, -500792236832, 235030416448, -89771423744, 27185833984, -6278031104, 1038269952, -109486080, 5529600).

Crossrefs

Cf. A064544, A065023, A065025, A065086.
Column k=9 of A288638.

Formula

G.f.: (2764800*x^35 -54743040*x^34 +535723776*x^33 -3484062592*x^32 +17047244288*x^31 -67056352000*x^30 +220043616032*x^29 -610136398384*x^28 +1428398369904*x^27 -2800237309450*x^26 +4555415187081*x^25 -6116515610358*x^24 +6790044899737*x^23 -6333177380214*x^22 +5196278284089*x^21 -4097957831766*x^20 +3395084470412*x^19 -2936902021347*x^18 +2431358755383*x^17 -1791957130479*x^16 +1141680065910*x^15 -626654334304*x^14 +298277671441*x^13 -124021600362*x^12 +45181016933*x^11 -14371192060*x^10 +3953830871*x^9 -928344574*x^8 +183129613*x^7 -29820446*x^6 +3925130*x^5 -406196*x^4 +31739*x^3 -1755*x^2 +61*x-1) / ((10*x-1) *(5*x-1) *(4*x-1)^2 *(3*x-1)^3 *(2*x-1)^8 *(x-1)^14). - Alois P. Heinz, Jun 12 2017
Limit_{n->oo} a(n)/10^n = 1/2. - Stefano Spezia, Sep 09 2023

A065023 Number of states in minimal automaton that recognizes biquanimous numbers in base n.

Original entry on oeis.org

2, 4, 10, 21, 51, 89, 203, 370, 715, 1197, 2418, 3813, 7175, 11379, 19026, 29809, 51618, 75378, 125951, 185025, 285449
Offset: 2

Views

Author

N. J. A. Sloane, Nov 03 2001

Keywords

Comments

A biquanimous number (A064544) is a number whose digits can be split into two groups with equal sums.

References

  • Jeffrey Shallit, A Second Course in Formal Languages and Automata Theory, Cambridge, 2008; see Exercise 3.55.
  • William P. Thurston, personal communication.

Links

Crossrefs

Cf. A064544, A065024, A065025.

Extensions

a(11)-a(16) from Jingzhe Tang, Mar 21 2018
a(17)-a(22) from Sean A. Irvine, Aug 08 2023

A288550 Number of strings of n digits from 1...9 such that a signed summation of the digits exists making the sum = 0.

Original entry on oeis.org

1, 0, 9, 108, 1569, 20230, 229203, 2278745, 21214753, 192899244, 1741242069, 15684465423, 141196229849, 1270871708340, 11438182427193, 102944790695746, 926507214592705, 8338579980466304, 75047276148618205, 675425698975426255, 6078832109331582297
Offset: 0

Views

Author

Hugo Pfoertner, Jun 11 2017

Keywords

Examples

			a(2)=9, because 11, 22, ..., 99 can be written as 1-1=0, 2-2=0, ...

Links

Crossrefs

Cf. A065024, A065025, A065086, A288351, A288352.

Formula

Limit_{n->oo} a(n)/9^n = 1/2.
G.f.: (4447872*x^35 +731808*x^34 -31561200*x^33 -9438744*x^32 +95630316*x^31 +43022340*x^30 -156898794*x^29 -98774388*x^28 +140941738*x^27 +120112934*x^26 -46571519*x^25 -49352408*x^24 -50794519*x^23 -70733352*x^22 +118351595*x^21 +120154070*x^20 -162641593*x^19 -54549200*x^18 +156403902*x^17 -38131997*x^16 -93427552*x^15 +56672934*x^14 +28535743*x^13 -26850890*x^12 -1996107*x^11 +5000082*x^10 -264871*x^9 -434046*x^8 +41593*x^7 +13610*x^6 +4622*x^5 -4524*x^4 +1500*x^3 -276*x^2 +26*x -1) / ((9*x-1) *(4*x-1) *(3*x-1)^2 *(2*x-1)^3 *(x+1)^7 *(x-1)^8). - Alois P. Heinz, Jun 11 2017
a(n) = (9^n - A065025(n))/2 for n>0. - Alois P. Heinz, Jun 12 2017

Extensions

a(11)-a(20) from Alois P. Heinz, Jun 11 2017

A065086 Number of n-digit biquanimous numbers in base 10 not allowing leading zeros.

Original entry on oeis.org

1, 9, 126, 1920, 27190, 347168, 3990467, 42744527, 440764556, 4464045276, 44863859589, 449488519847, 4498059105204, 44992451829730, 449969622539954, 4499873022468708, 44999449703306768, 449997540235466340, 4499988731927483569, 44999947410278947807
Offset: 1

Views

Author

David W. Wilson, Nov 07 2001

Keywords

Comments

A biquanimous number (A064544) is a number whose digits can be split into two groups with equal sums.

Examples

			a(1) = 1 since 0 is the only 1-digit biquam. a(2) = 9 because there are 9 2-digit biquams, namely 11, 22, 33, 44, 55, 66, 77, 88 and 99.

References

Crossrefs

Cf. A064544, A065023, A065025, A065024.
Showing 1-5 of 5 results.

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

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