A288638 -id:A288638 - 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

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733

Offset: 1

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

[(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017

A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)

a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

A288687 Number of n-digit biquanimous strings using digits {0,1,2,3}.

Original entry on oeis.org

1, 1, 4, 19, 92, 421, 1830, 7687, 31624, 128521, 518666, 2084875, 8361996, 33497101, 134094862, 536608783, 2146926608, 8588754961, 34357248018, 137433710611, 549744803860, 2199000186901, 8796044787734, 35184271425559, 140737278640152, 562949517213721

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-37,64,-52,16).

Column k=3 of A288638.

LinearRecurrence[{10,-37,64,-52,16},{1,1,4,19,92,421},30] (* Harvey P. Dale, Jul 29 2017 *)

Vec((1 - 9*x + 31*x^2 - 48*x^3 + 38*x^4 - 16*x^5) / ((1 - x)^2*(1 - 2*x)^2*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Dec 16 2017

G.f.: (1 - 9*x + 31*x^2 - 48*x^3 + 38*x^4 - 16*x^5) / ((1 - x)^2*(1 - 2*x)^2*(1 - 4*x)).
a(n) = 1 + A064671(n) for n > 0.
From Colin Barker, Dec 16 2017: (Start)
a(n) = (2^(2*n-1) + n - 2^(n-1)*(1+n)).
a(n) = 10*a(n-1) - 37*a(n-2) + 64*a(n-3) - 52*a(n-4) + 16*a(n-5) for n>5.
(End)

A288688 Number of n-digit biquanimous strings using digits {0,1,...,4}.

Original entry on oeis.org

1, 1, 5, 31, 201, 1206, 6751, 36051, 187025, 954136, 4822527, 24251877, 121631329, 609151986, 3048441935, 15249510871, 76267672545, 381394509228, 1907131201327, 9536109476745, 47681856564305, 238413094649734, 1192076649237855, 5960416195898811

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18, -137, 582, -1527, 2574, -2795, 1890, -724, 120).

Column k=4 of A288638.

G.f.: -(60*x^10 -422*x^9 +1044*x^8 -1855*x^7 +2394*x^6 -2122*x^5 +1273*x^4 -504*x^3 +124*x^2 -17*x +1) / ((5*x-1) *(3*x-1) *(2*x-1)^3 *(x-1)^4).

A288689 Number of n-digit biquanimous strings using digits {0,1,...,5}.

Original entry on oeis.org

1, 1, 6, 46, 376, 2841, 19718, 128535, 805848, 4942711, 29970542, 180700389, 1086570460, 6525662885, 39170135870, 235062159691, 1410477973872, 8463133736523, 50779476069198, 304678570340665, 1828075815690100, 10968466276145161, 65810827526263678

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (23, -231, 1351, -5153, 13557, -25301, 33829, -32226, 21368, -9376, 2448, -288).

Column k=5 of A288638.

CoefficientList[Series[(144x^14-1224x^13+4976x^12-11002x^11+18115x^10-25255x^9+30319x^8-29516x^7+21747x^6-11691x^5+4506x^4-1212x^3+214x^2-22x+1)/((6x-1)(3x-1)(2x-1)^4(x-1)^6),{x,0,40}],x] (* or *) LinearRecurrence[{23,-231,1351,-5153,13557,-25301,33829,-32226,21368,-9376,2448,-288},{1,1,6,46,376,2841,19718,128535,805848,4942711,29970542,180700389,1086570460,6525662885,39170135870},40] (* Harvey P. Dale, Aug 18 2025 *)

G.f.: (144*x^14 -1224*x^13 +4976*x^12 -11002*x^11 +18115*x^10 -25255*x^9 +30319*x^8 -29516*x^7 +21747*x^6 -11691*x^5 +4506*x^4 -1212*x^3 +214*x^2 -22*x +1) / ((6*x-1) *(3*x-1) *(2*x-1)^4 *(x-1)^6).

A288690 Number of n-digit biquanimous strings using digits {0,1,...,6}.

Original entry on oeis.org

1, 1, 7, 64, 633, 5801, 48245, 372345, 2743793, 19706380, 139666975, 983424751, 6902981425, 48383824035, 338898209049, 2373012819041, 16613639684833, 116304663546706, 814166474511867, 5699292259116239, 39895529322328145, 279270568611716769, 1954901225698086549

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (35, -557, 5373, -35284, 167758, -599018, 1643178, -3510009, 5878447, -7726457, 7932089, -6287970, 3771376, -1654048, 500240, -93216, 8064).

Column k=6 of A288638.

G.f.: -(4032*x^20 -50640*x^19 +300760*x^18 -1131816*x^17 +2828844*x^16 -5169218*x^15 +7372355*x^14 -8685305*x^13 +8984178*x^12 -8471674*x^11 +7172188*x^10 -5192457*x^9 +3073627*x^8 -1446472*x^7 +532167*x^6 -150791*x^5 +32203*x^4 -4997*x^3 +529*x^2 -34*x +1) / ((7*x-1) *(4*x-1) *(3*x-1)^2 *(2*x-1)^5 *(x-1)^8).

A288691 Number of n-digit biquanimous strings using digits {0,1,...,7}.

Original entry on oeis.org

1, 1, 8, 85, 988, 10696, 104676, 939863, 7980376, 65679175, 532115106, 4279710436, 34311898336, 274729437763, 2198561705222, 17590732383423, 140732804110800, 1125884352801489, 9007145852828686, 72057404552774828, 576460059887430848, 4611683426113781209

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (40, -734, 8256, -64063, 365480, -1593840, 5449456, -14855999, 32647032, -58206694, 84416480, -99491505, 94840120, -72455660, 43708272, -20343280, 7044608, -1708352, 258816, -18432).

Column k=7 of A288638.

G.f.: (9216*x^24 -129408*x^23+881824*x^22 -3910528*x^21 +12761816*x^20 -31636760*x^19 +62309010*x^18 -101187417*x^17 +139600728*x^16 -166854657*x^15 +174047204*x^14 -158809357*x^13 +127270681*x^12 -89880736*x^11 +55637484*x^10 -29668152*x^9 +13297571*x^8 -4887342*x^7 +1441132*x^6 -333899*x^5 +59267*x^4 -7757*x^3 +702*x^2 -39*x +1) / ((8*x-1) *(4*x-1) *(3*x-1)^2 *(2*x-1)^6 *(x-1)^10).

A288692 Number of n-digit biquanimous strings using digits {0,1,...,8}.

Original entry on oeis.org

1, 1, 9, 109, 1457, 18231, 205837, 2112384, 20341201, 189013501, 1725377947, 15622028115, 140950202021, 1269874954518, 11433967343409, 102926130336136, 926421241593985, 8338171770377449, 75045297684511343, 675415983886759045, 6078784024809901485

Offset: 0

Alois P. Heinz, Jun 13 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (49, -1114, 15704, -154592, 1133720, -6448172, 29219972, -107498498, 325391618, -818066864, 1719218324, -3031985264, 4494756128, -5598534740, 5843651372, -5085390293, 3660194213, -2153194198, 1017730084, -376961752, 105342192, -20876832, 2614464, -155520).

Column k=8 of A288638.

G.f.: -(77760*x^29 -1384992*x^28 +12134448*x^27 -69723432*x^26 +294884564*x^25 -975139270*x^24 +2569875380*x^23 -5484438559*x^22 +9604318227*x^21 -13997308104*x^20 +17318565226*x^19 -18674789778*x^18 +18024625668*x^17 -15836546206*x^16 +12680214123*x^15 -9166750277*x^14 +5925087725*x^13 -3404858524*x^12 +1731743202*x^11 -773011544*x^10 +298459228*x^9 -97814317*x^8 +26669775*x^7 -5929660*x^6 +1052200*x^5 -145030*x^4 +14922*x^3 -1074*x^2 +48*x -1) / ((9*x-1) *(5*x-1) *(3*x-1)^3 *(2*x-1)^7 *(x-1)^12).

A288693 Number of n-digit biquanimous strings using digits {0,1,...,n}.

Original entry on oeis.org

1, 1, 3, 19, 201, 2841, 48245, 939863, 20341201, 487875964, 12830282835, 370205055144, 11629998323185, 396693714869323, 14593231979817751, 576427808563042857

Offset: 0

Alois P. Heinz, Jun 13 2017

nonn

A biquanimous string is a string whose digits can be split into two groups with equal sums.

			a(2) = 3: 00, 11, 22.
a(3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.

Main diagonal of A288638.

b:= proc(n, k, s) option remember;
      `if`(n=0, `if`(s={}, 0, 1), add(b(n-1, k, select(y->
       y<=(n-1)*k, map(x-> [abs(x-i), x+i][], s))), i=0..k))
    end:
a:= n-> b(n$2, {0}):
seq(a(n), n=0..10);

b[n_, k_, s_] := b[n, k, s] = If[n == 0, If[s == {}, 0, 1], Sum[b[n-1, k, Select[Flatten[{Abs[#-i], #+i}& /@ s], # <= (n-1)*k&]], {i, 0, k}]];
a[n_] := b[n, n, {0}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* Jean-François Alcover, May 17 2022, after Alois P. Heinz *)

cp's OEIS Frontend

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

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A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

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A288687 Number of n-digit biquanimous strings using digits {0,1,2,3}.

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A288688 Number of n-digit biquanimous strings using digits {0,1,...,4}.

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A288689 Number of n-digit biquanimous strings using digits {0,1,...,5}.

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A288690 Number of n-digit biquanimous strings using digits {0,1,...,6}.

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A288691 Number of n-digit biquanimous strings using digits {0,1,...,7}.

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A288692 Number of n-digit biquanimous strings using digits {0,1,...,8}.

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A288693 Number of n-digit biquanimous strings using digits {0,1,...,n}.

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