A197083 -id:A197083 - cp's OEIS frontend

A240927 Positive integers with 2k digits (the first of which is not 0) where the sum of the first k digits equals the sum of the last k digits.

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1010, 1102, 1111, 1120, 1203, 1212, 1221, 1230, 1304, 1313, 1322, 1331, 1340, 1405, 1414, 1423, 1432, 1441, 1450, 1506, 1515, 1524, 1533, 1542, 1551, 1560, 1607, 1616, 1625, 1634, 1643, 1652, 1661, 1670, 1708, 1717

Offset: 1

Martin Renner, Aug 03 2014

These integers are sometimes called balanced numbers.

There are 9, 615, 50412, 4379055, 392406145, ... 2k-digit balanced numbers with k >= 1.

			1423 is a 4-digit balanced number, because the sum of the first 2 digits equals the sum of the last 2 digits: 1 + 4 = 2 + 3.

Cambridge Colleges Sixth Term Examination Papers (STEP) 2007, Paper I, Section A (Pure Mathematics), Nr. 1.

Harvey P. Dale, Table of n, a(n) for n = 1..624 (All terms through 9999)

Cf. A016061, A197083, A240928, A240929.

sfslQ[n_]:=Module[{id=IntegerDigits[n],len},len=Length[id]/2;Total[Take[ id,len]]==Total[Take[id,-len]]]; Select[Table[Range[10^n,10^(n+1)-1],{n,1,3,2}]// Flatten,sfslQ] (* Harvey P. Dale, Jun 24 2020 *)

A240929 Number of 10-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

Original entry on oeis.org

126, 6046, 88428, 694360, 3705741, 15192604, 51418473, 150420187, 392406145, 933294637, 2056947827, 4253047045, 8329101326, 15566783605, 27934647638, 48371293570, 81155221112, 132379936520, 210555362990, 327359243694, 498565022483, 745175639274, 1094795785319

Offset: 2

Martin Renner, Aug 03 2014

These integers are sometimes called balanced numbers.

Cambridge Colleges Sixth Term Examination Papers (STEP) 2007, Paper I, Section A (Pure Mathematics), Nr. 1.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016061, A197083, A240927, A240928.

def A240929(n): return n*(n*(n*(n*(n*(n*(n*(n*(156190*n-140571)+29400)-30870)+3990)-8379)-3100)-1620)-5040)//362880 # Chai Wah Wu, May 08 2024

a(n) = n*(n-1)*(156190*n^7 + 15619*n^6 + 45019*n^5 + 14149*n^4 + 18139*n^3 + 9760*n^2 + 6660*n + 5040)/362880
From Chai Wah Wu, May 08 2024: (Start)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 11.
G.f.: x^2*(x^7 + 326*x^6 + 7942*x^5 + 42341*x^4 + 67030*x^3 + 33638*x^2 + 4786*x + 126)/(x - 1)^10. (End)

A240928 Number of 8-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

Original entry on oeis.org

35, 750, 6174, 31025, 114961, 346193, 896876, 2072694, 4379055, 8606312, 15936426, 28073487, 47400509, 77164915, 121695128, 186650684, 279308283, 408886194, 586909430, 827618109, 1148421417, 1570399589, 2118856324, 2823924050, 3721224455, 4852586700

Offset: 2

Martin Renner, Aug 03 2014

These integers are sometimes called balanced numbers.

Cambridge Colleges Sixth Term Examination Papers (STEP) 2007, Paper I, Section A (Pure Mathematics), Nr. 1.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016061, A197083, A240927, A240929.

Table[n(n-1)(1208n^5+151n^4+291n^3+116n^2+88n+60)/2520,{n,2,40}] (* Harvey P. Dale, Mar 18 2022 *)

a(n) = n*(n-1)*(1208*n^5+151*n^4+291*n^3+116*n^2+88*n+60)/2520.

G.f.: x^2*(x^5+83*x^4+673*x^3+1154*x^2+470*x+35)/(x-1)^8. - Alois P. Heinz, Mar 24 2022

A179441 Number of solutions to a+b+c < d+e with each of a,b,c,d,e in {1..n+1}.

Original entry on oeis.org

1, 21, 121, 432, 1182, 2723, 5558, 10368, 18039, 29689, 46695, 70720, 103740, 148071, 206396, 281792, 377757, 498237, 647653, 830928, 1053514, 1321419, 1641234, 2020160, 2466035, 2987361, 3593331, 4293856, 5099592, 6021967, 7073208, 8266368, 9615353, 11134949, 12840849

Offset: 1

Bobby Milazzo, Jul 14 2010

			a(1) = 1 since from {1,2} there is only one solution: {1,1,1} for a,b,c and {2,2} for d,e.
a(2) = 21 since from {1,2,3} there are 21 ways to select a,b,c,d,e such that a+b+c < d+e.

Mathematics and Computer Education 1988 - 89 #261 Unsolved.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A197083.

k=10;
Table[p=Expand[Sum[x^k,{k,1,n}]^2 Sum[1/x^k,{k,1,n}]^3];
Twowins=Drop[CoefficientList[p,x],1]//Total,{n,2,k}]

a(n)=(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n)/120 \\ Andrew Howroyd, Apr 15 2021

a(n) = (1/120)*(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.
G.f.: x*(1 + 15*x + 10*x^2 + x^3)/(1 - x)^6.

Name edited and terms a(24) and beyond from Andrew Howroyd, Apr 15 2021

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A240927 Positive integers with 2k digits (the first of which is not 0) where the sum of the first k digits equals the sum of the last k digits.

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A240929 Number of 10-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

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A240928 Number of 8-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

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A179441 Number of solutions to a+b+c < d+e with each of a,b,c,d,e in {1..n+1}.

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