A055436 -id:A055436 - cp's OEIS frontend

A053061 a(n) is the decimal concatenation of n and n^2.

11, 24, 39, 416, 525, 636, 749, 864, 981, 10100, 11121, 12144, 13169, 14196, 15225, 16256, 17289, 18324, 19361, 20400, 21441, 22484, 23529, 24576, 25625, 26676, 27729, 28784, 29841, 30900, 31961, 321024, 331089, 341156, 351225, 361296, 371369, 381444, 391521

Offset: 1

Felice Russo, Feb 25 2000

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Georg Fischer, Table of n, a(n) for n = 1..10000

Cf. A020338, A055436, A061086, A175605, A253445.

[Seqint(Intseq(n^2) cat Intseq(n)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n^2]]],{n,40}] (* Harvey P. Dale, May 24 2012 *)

def a(n): return int(str(n) + str(n*n))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Nov 24 2021

a(n) = n*(10^floor(2*log_10(n) + 1) + n). - Henry Bottomley, May 18 2000

a(n) ~ n^3. - Charles R Greathouse IV, Sep 19 2012

More terms from James Sellers, Feb 28 2000

A055438 a(n) = 100*n^2 + n.

Original entry on oeis.org

101, 402, 903, 1604, 2505, 3606, 4907, 6408, 8109, 10010, 12111, 14412, 16913, 19614, 22515, 25616, 28917, 32418, 36119, 40020, 44121, 48422, 52923, 57624, 62525, 67626, 72927, 78428, 84129, 90030, 96131, 102432, 108933, 115634, 122535

Offset: 1

Henry Bottomley, May 18 2000

The identity (200n+1)^2 - (100n^2+n)*20^2 = 1 can be written as A157956(n)^2 - a(n)*20^2 = 1 (see Barbeau's paper). Also, the identity (80000n^2 + 800n + 1)^2 - (100n^2 + n)*(8000n + 40)^2 = 1 can be written as A157664(n)^2 - a(n)*A157663(n)^2 = 1 (see the comment from Bruno Berselli in A157664). - Vincenzo Librandi, Feb 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 14 in the first table at p. 85, case d(t) = t*(10^2*t+1)).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A157956, A157663, A157664, A002378, A055437; a(n) = A055436(n) if 10 <= n < 100.

Different from A031698.

I:=[101, 402, 903]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 04 2012

LinearRecurrence[{3, -3, 1}, {101, 402, 903}, 50] (* Vincenzo Librandi, Feb 04 2012 *)
Table[100n^2+n,{n,40}] (* Harvey P. Dale, May 15 2018 *)

for(n=1, 50, print1(100*n^2+n", ")); \\ Vincenzo Librandi, Feb 04 2012

G.f.: x*(-101-99*x)/(x-1)^3. - Vincenzo Librandi, Feb 04 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 04 2012

A055437 a(n) = 10*n^2+n.

Original entry on oeis.org

11, 42, 93, 164, 255, 366, 497, 648, 819, 1010, 1221, 1452, 1703, 1974, 2265, 2576, 2907, 3258, 3629, 4020, 4431, 4862, 5313, 5784, 6275, 6786, 7317, 7868, 8439, 9030, 9641, 10272, 10923, 11594, 12285, 12996, 13727, 14478, 15249, 16040, 16851, 17682, 18533

Offset: 1

Henry Bottomley, May 18 2000

a(n) = A055436(n) if 1<=n<10.

Number of edges in the join of the complete 4-partite graph of order 4n and the cycle graph of order n, K_n,n,n,n * C_n. - Roberto E. Martinez II, Jan 07 2002

			From the third formula: a(8) = 648 = 16^2 -17^2 +18^2 ... +30^2 -31^2 +32^2 = -33^2 +34^2 -35^2 ... +46^2 -47^2 +48^2.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A002378, A055436, A055438.

[10*n^2+n : n in [1..50]]; // Wesley Ivan Hurt, Apr 27 2016

A055437:=n->10*n^2+n: seq(A055437(n), n=1..50); # Wesley Ivan Hurt, Apr 27 2016

Table[10 n^2 + n, {n, 50}] (* Wesley Ivan Hurt, Apr 27 2016 *)

a(n)=10*n^2+n \\ Charles R Greathouse IV, Jun 17 2017

From Bruno Berselli, Nov 26 2013: (Start)
G.f.: x*(11 + 9*x) / (1 - x)^3.
a(n) = Sum_{i=0..2*n} (-1)^i*(2*n+i)^2.
a(n) = Sum_{i=1..2*n} (-1)^i*(4*n+i)^2. (End)
From Wesley Ivan Hurt, Apr 27 2016: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.
a(n) = (1/5) * Sum_{i=0..10*n} i. (End)
E.g.f.: x*(11 + 10*x)*exp(x). - Ilya Gutkovskiy, Apr 27 2016
a(n) = A000217(6*n) - A000217(4*n). - Bruno Berselli, Sep 21 2016

A239459 Concatenation of n^3 and n.

Original entry on oeis.org

11, 82, 273, 644, 1255, 2166, 3437, 5128, 7299, 100010, 133111, 172812, 219713, 274414, 337515, 409616, 491317, 583218, 685919, 800020, 926121, 1064822, 1216723, 1382424, 1562525, 1757626, 1968327, 2195228, 2438929, 2700030, 2979131, 3276832, 3593733

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n.

			a(13) = 219713 because 2197 = 13^3 and 13 = 13^1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A239460, A239461.

Cf. A061087, A239462, A239463, A239464.

Table[ToExpression[ToString[n^3]<>ToString[n]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[n^3*10^IntegerLength[n]+n,{n,40}] (* Harvey P. Dale, Sep 07 2020 *)

PARI
```
vector(100, n, eval(Str(n^3, n)))
```

A239460 Concatenation of n^2 and n^3.

Original entry on oeis.org

11, 48, 927, 1664, 25125, 36216, 49343, 64512, 81729, 1001000, 1211331, 1441728, 1692197, 1962744, 2253375, 2564096, 2894913, 3245832, 3616859, 4008000, 4419261, 48410648, 52912167, 57613824, 62515625, 67617576, 72919683, 78421952, 84124389, 90027000

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n^2.

			a(9) = 81729 because 81 = 9^2 and 729 = 9^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A061087, A239459, A239461, A239462, A239463, A239464.

Table[ToExpression[ToString[n^2]<>ToString[n^3]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[FromDigits[Join[Flatten[IntegerDigits[{n^2,n^3}]]]],{n,30}] (* Harvey P. Dale, Oct 27 2019 *)

PARI
```
vector(100, n, eval(Str(n^2, n^3)))
```

A239461 Concatenation of n^3 and n^2.

Original entry on oeis.org

11, 84, 279, 6416, 12525, 21636, 34349, 51264, 72981, 1000100, 1331121, 1728144, 2197169, 2744196, 3375225, 4096256, 4913289, 5832324, 6859361, 8000400, 9261441, 10648484, 12167529, 13824576, 15625625, 17576676, 19683729, 21952784, 24389841, 27000900

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n^2.

			a(9) = 72981 because 729 = 9^3 and 81 = 9^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A239459, A239460.

Cf. A061087, A239462, A239463, A239464.

Table[ToExpression[ToString[n^3]<>ToString[n^2]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[FromDigits[Flatten[IntegerDigits/@{n^3,n^2}]],{n,30}] (* Harvey P. Dale, Nov 28 2024 *)

PARI
```
vector(100, n, eval(Str(n^3, n^2)))
```

A274620 If n^2 has an even number of digits, write n after the left half of the digits of n^2 and before the right half, otherwise if n^2 has 2t+1 digits, write n after the first t digits of n^2 and before the last t+1 digits.

Original entry on oeis.org

11, 24, 39, 146, 255, 366, 479, 684, 891, 11000, 11121, 11244, 11369, 11496, 21525, 21656, 21789, 31824, 31961, 42000, 42141, 42284, 52329, 52476, 62525, 62676, 72729, 72884, 82941, 93000, 93161, 103224, 103389, 113456, 123525, 123696, 133769, 143844, 153921, 164000

Offset: 1

N. J. A. Sloane, Jul 03 2016

In short, write n in the middle of n^2.

Portions of this sequence are sometimes given as puzzles.

			4^2 = 16 so a(4) = 1.4.6 = 146.
19^2 = 361 so a(19) = 3.19.61 = 31961.

J. A. Reeds, Personal communication to N. J. A. Sloane, Jun 04 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A053061, A055436.

nterms=100;Table[FromDigits[Flatten[Insert[d=IntegerDigits[n^2],IntegerDigits[n],Floor[Length[d]/2]+1]]],{n,nterms}] (* Paolo Xausa, Nov 24 2021 *)

def a(n):
    ss = str(n*n)
    t = len(ss)//2
    return int(ss[:t] + str(n) + ss[t:])
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Nov 24 2021

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35, 38, 40, 41, 45, 46, 50, 55, 56, 60, 63, 64, 65, 66, 68, 70, 74, 75, 76, 77, 80, 81, 83, 85, 88, 90, 91, 95, 96, 97, 99, 100, 101, 102, 105, 109, 110

Offset: 1

Jovan Radenkovicc, Nov 15 2024

Problem: Is there a real constant c such that a(n) < n^c for all positive integers n?

All of A136808, A136809, A136816, ..., A137079 are subsequences. Many if not most terms of A058411, A058413, ... ("tridigital solutions") are also in this sequence; see also Hisanori Mishima's web page for some nontrivial solutions. - M. F. Hasler, Feb 02 2025

			816 is in the sequence since 816^2 = 665856 and both together use at most 4 distinct digits.
149 is not in the sequence since 149^2 = 22201 and both together use 5 distinct digits.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 748 terms from Jovan Radenkovicc)
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
Hisanori Mishima, Sporadic tridigital solutions
OEIS index for squares having only given digits

Supersequence of A136808, A136809, A136816, A136822.

Cf. A043537, A053061, A055436, A154155.

[n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];

Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)

isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024

is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)),s)<5 \\ Charles R Greathouse IV, Jan 30 2025

is1(n)=#setunion(Set(digits(n^2)),Set(digits(n)))<5
ok(m)=my(d=concat(apply(k->digits(lift(k)), [m,m^2]))
test(d)=my(v=List(),D=10^d); for(n=0,D-1, if(ok(Mod(n,D)), listput(v,n))); Vec(v)
res=test(8); \\ build a list of residues mod 10^8
D=diff(concat(res,res[1]+10^8)); #D
u=List(); for(n=0,10^7, if(is1(n) && !setsearch(n,res), listput(u,n))); \\ build exceptions
setminus(select(is1,[0..n]),list(n))
list(lim)=my(v=List(u)); forstep(n=0,lim,D, if(is1(n), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 30 2025

def ok(n): return len(set(str(n)+str(n**2))) <= 4
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Nov 18 2024

A043537(A053061(a(n))) <= 4.

Trivially, a(n) >> n^1.66... where the exponent is log(10)/log(4) (A154155). - Charles R Greathouse IV, Jan 30 2025

A055435 (1/18)*Difference between concatenation of n and n^2 and concatenation of n^2 and n.

Original entry on oeis.org

0, -1, -3, 14, 15, 15, 14, 12, 9, 5, -55, -126, -208, -301, -405, -520, -646, -783, -931, -1090, -1260, -1441, -1633, -1836, -2050, -2275, -2511, -2758, -3016, -3285, -3565, 12144, 12342, 12529, 12705, 12870, 13024, 13167, 13299, 13420

Offset: 1

Henry Bottomley, May 18 2000

Table[((n*10^IntegerLength[x = n^2] + x) - (n^2*10^IntegerLength[n] + n))/18, {n, 40}] (* Jayanta Basu, Jul 12 2013 *)

a(n) = (A053061(n) - A055436(n))/18 = n*(10^floor(2*log_10(W2)+1) - 1 - n*(10^floor(log_10(W2)+1)-1))/18.

cp's OEIS Frontend

A053061 a(n) is the decimal concatenation of n and n^2.

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A055438 a(n) = 100*n^2 + n.

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A055437 a(n) = 10*n^2+n.

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A239459 Concatenation of n^3 and n.

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A239460 Concatenation of n^2 and n^3.

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A239461 Concatenation of n^3 and n^2.

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A274620 If n^2 has an even number of digits, write n after the left half of the digits of n^2 and before the right half, otherwise if n^2 has 2t+1 digits, write n after the first t digits of n^2 and before the last t+1 digits.

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