A064544 -id:A064544 - cp's OEIS frontend

A288638 Number A(n,k) of n-digit biquanimous strings using digits {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 33, 16, 1, 1, 1, 6, 31, 92, 106, 32, 1, 1, 1, 7, 46, 201, 421, 333, 64, 1, 1, 1, 8, 64, 376, 1206, 1830, 1030, 128, 1, 1, 1, 9, 85, 633, 2841, 6751, 7687, 3153, 256, 1

Offset: 0

Alois P. Heinz, Jun 12 2017

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

			A(2,2) = 3: 00, 11, 22.
A(3,2) = 10: 000, 011, 022, 101, 110, 112, 121, 202, 211, 220.
A(3,3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.
A(4,1) = 8: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111.
Square array A(n,k) begins:
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  2,    3,    4,     5,      6,      7,      8, ...
  1,  4,   10,   19,    31,     46,     64,     85, ...
  1,  8,   33,   92,   201,    376,    633,    988, ...
  1, 16,  106,  421,  1206,   2841,   5801,  10696, ...
  1, 32,  333, 1830,  6751,  19718,  48245, 104676, ...
  1, 64, 1030, 7687, 36051, 128535, 372345, 939863, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns k=0-9 give: A000012, A011782, A053156, A288687, A288688, A288689, A288690, A288691, A288692, A065024.
Rows n=0+1,2-3 give: A000012, A000027(k+1), A005448(k+1).
Main diagonal gives A288693.
Cf. A064544, A064914.

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, k)-> b(n, k, {0}):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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_, k_] := b[n, k, {0}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 08 2018, from Maple *)

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

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

A065025 Consider biquanimous numbers that exclude 0's; sequence gives number of n-digit non-biquanimous numbers - number of n-digit biquanimous numbers.

Original entry on oeis.org

9, 63, 513, 3423, 18589, 73035, 225479, 617215, 1622001, 4300263, 12128763, 37076783, 122411649, 427600575, 1550703157, 5759666431, 21738733961, 82999762711, 319722139579, 1240393764207, 4840363237201, 18979321319087, 74713018378209, 295061102101311

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.
William P. Thurston, Biquanimous Generating Function, math-fun mailing list, November 2001.

Alois P. Heinz, Table of n, a(n) for n = 1..1660
Index entries for linear recurrences with constant coefficients, signature (17, -114, 348, -228, -1524, 3888, -216, -11046, 11382, 12012, -26544, 84, 28812, -13152, -15816, 13407, 3201, -5834, 628, 984, -288).

Cf. A064544, A065023, A065024, A288550.

From Alois P. Heinz, Jun 12 2017: (Start)
G.f.: -x*(988416*x^33 +272448*x^32 -6983328*x^31 -2873424*x^30 +20931912*x^29 +11886288*x^28 -33545700*x^27 -25677164*x^26 +28467368*x^25 +29854804*x^24 -7032026*x^23 -11748538*x^22 -12593064*x^21 -17118040*x^20 +24399398*x^19 +29412358*x^18 -32880510*x^17 -15770937*x^16 +33016792*x^15 -4824040*x^14 -21307320*x^13 +10258240*x^12 +7474762*x^11 -5162898*x^10 -999324*x^9 +1008806*x^8 +39654*x^7 -89810*x^6 +3200*x^5 +992*x^4 +1248*x^3 -468*x^2 +90*x -9) / ((4*x-1) *(3*x-1)^2 *(2*x-1)^3 *(x+1)^7 *(x-1)^8).
a(n) = 9^n - 2 * A288550(n). (End)

New offset and 4 more terms from Alois P. Heinz, Jun 11 2017

A064686 a(n) = number of n-digit base-3 biquams.

Original entry on oeis.org

0, 2, 7, 23, 73, 227, 697, 2123, 6433, 19427, 58537, 176123, 529393, 1590227, 4774777, 14332523, 43013953, 129074627, 387289417, 1161999323, 3486260113, 10459304627, 31378962457, 94138984523, 282421147873, 847271832227

Offset: 1

David W. Wilson, Oct 10 2001

A biquam or biquanimous number (A064544) is a number whose digits can be split into two groups with equal sum.
This is the same as A083313 (apart from the initial term). Proof: Let sum(w) denote the sum of the digits of w. There are 2*3^(n-1) n-digit base-3 numbers: w = (w_1,w_2,...,w_n) with w_i in {0,1,2} for all i and w_1 != 0. Partition them into 4 classes: (i) sum(w) is odd, (ii) sum(w) is even, w contains no 1's and has an odd number of 2s, (iii) sum(w) is even, w contains no 1's and has an even number of 2s and (iv) sum(w) is even and w contains some 1's. Clearly, no biquams occur in cases (i) and (ii), case (iii) consists entirely of biquams and, we claim, so does case (iv). For case (iv) forces an even number, say 2k, of 1's. An even number of 2s clearly gives a biquam and an odd number 2m+1 of 2s does too because {m 2s, (k+1) 1's} and {(m+1) 2s, (k-1) 1's} is a biquam split. There are 3^(n-1) w's in case (i) and 2^(n-2) w's in case (ii) and hence 2*3^(n-1) - (3^(n-1) + 2^(n-2)) = 3^(n-1) - 2^(n-2) (A083313) biquams among n-digit base-3 numbers. - David Callan, Sep 15 2004
a(n) % 100 = 23  for n = 4*k-1, k>=1; a(n) % 100 = 27  for n = 4*k+1, k>=1. - Alex Ratushnyak, Jul 03 2012
The fraction of biquams for any base approaches 1/2 as the number of digits grows but only if you count leading zeros. Without counting leading zeros, the fraction appears to converge to (b-1)/2b where b is the base used. For base 3 this is 1/3 which fits the data in this sequence (see paper cited below for proofs and the OEIS data collated as fractions). - Timothy Varghese, Aug 08 2021

Timothy Varghese and George Varghese, The Stash-Repair Method applied to Partitionable Numbers,
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Essentially the same as A083313.

Cf. A053152 (partial sums).

print([0]+[3**n - 2**(n-1) for n in range(1,29)])
# Alex Ratushnyak, Jul 02 2012

a(1) = 0, a(n) = 3^(n-1)-2^(n-2) for n>=2. - Alex Ratushnyak, Jul 02 2012

a(n) = 5*a(n-1)-6*a(n-2) for n>3. G.f.: -x^2*(3*x-2) / ((2*x-1)*(3*x-1)). - Colin Barker, May 27 2013

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

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.

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

Jingzhe Tang, C++ code calculating terms

Cf. A064544, A065024, A065025.

a(11)-a(16) from Jingzhe Tang, Mar 21 2018

a(17)-a(22) from Sean A. Irvine, Aug 08 2023

A064671 Number of n-digit base 4 biquanimous numbers (with leading 0's allowed, but not all-0 string).

Original entry on oeis.org

0, 3, 18, 91, 420, 1829, 7686, 31623, 128520, 518665, 2084874, 8361995, 33497100, 134094861, 536608782, 2146926607, 8588754960, 34357248017, 137433710610, 549744803859, 2199000186900, 8796044787733, 35184271425558, 140737278640151, 562949517213720

Offset: 1

John W. Layman, Oct 09 2001

A number is biquanimous (A064544) if its digits can be split into two groups with the same sum. - David W. Wilson, SeqFan memo, Oct 08 2001.

Empirical g.f.: x^2*(3 - 12*x + 22*x^2 - 16*x^3) / ((1 - x)^2*(1 - 2*x)^2*(1 - 4*x)). [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]
Conjectures from Colin Barker, Dec 16 2017: (Start)
a(n) = (2^n-2) * (1+2^n-n) / 2.
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)

More terms from Christian G. Bower, Oct 12 2001

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

David W. Wilson, Nov 07 2001

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

			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.

William P. Thurston, personal communication.

Cf. A064544, A065023, A065025, A065024.

A065571 Numbers whose decimal digits can be permuted to give a multiple of 11.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 139, 143, 145, 148, 154, 156, 157, 165, 166, 167, 175, 176, 178, 184, 187, 189, 193, 198, 202, 209, 211, 213, 220, 224, 231, 235, 242, 246, 249, 253, 257, 258, 264, 267, 268, 275, 276

Offset: 0

Reinhard Zumkeller, Dec 02 2001

a(k) = A064544(k) for k <= 263, a(263) = 1111. But the digits of the next biquam 1113 cannot be arranged to a multiple of 11. So we have a subset of those biquams, whose separating sets differ at most by 1 in size.

Select[Range[0,300],AnyTrue[FromDigits/@Permutations[ IntegerDigits[ #]], Divisible[ #,11]&]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

cp's OEIS Frontend

A288638 Number A(n,k) of n-digit biquanimous strings using digits {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A065025 Consider biquanimous numbers that exclude 0's; sequence gives number of n-digit non-biquanimous numbers - number of n-digit biquanimous numbers.

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A064686 a(n) = number of n-digit base-3 biquams.

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

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A064671 Number of n-digit base 4 biquanimous numbers (with leading 0's allowed, but not all-0 string).

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

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A065571 Numbers whose decimal digits can be permuted to give a multiple of 11.

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