A065086 -id:A065086 - cp's OEIS frontend

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

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

N. J. A. Sloane, Nov 03 2001

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

William P. Thurston, personal communication.

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).

Cf. A064544, A065023, A065025, A065086.

Column k=9 of A288638.

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

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

Hugo Pfoertner, Jun 11 2017

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

Alois P. Heinz, Table of n, a(n) for n = 0..1048
Index entries for linear recurrences with constant coefficients, signature (26, -267, 1374, -3360, 528, 17604, -35208, -9102, 110796, -90426, -134652, 238980, 28056, -272460, 102552, 155751, -117462, -34643, 53134, -4668, -9144, 2592).

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

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

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

cp's OEIS Frontend

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

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