A115273 -id:A115273 - cp's OEIS frontend

A115274 a(n) = n + A115273(n), where A115273(n) = 0 for n = 1..3.

1, 2, 3, 5, 7, 6, 9, 12, 9, 13, 17, 12, 17, 22, 15, 21, 27, 18, 25, 32, 21, 29, 37, 24, 33, 42, 27, 37, 47, 30, 41, 52, 33, 45, 57, 36, 49, 62, 39, 53, 67, 42, 57, 72, 45, 61, 77, 48, 65, 82, 51, 69, 87, 54, 73, 92, 57, 77, 97, 60, 81, 102, 63, 85, 107, 66, 89, 112, 69, 93, 117

Offset: 1

Zak Seidov, Jan 18 2006

Three arithmetic progressions interlaced: a(1..3) = 1..3 and d = a(n+3)-a(n) = 4,5,3.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A115273.

seq(op([1+4*j,2+5*j,3+3*j]),j=0..100); # Robert Israel, May 11 2015

Table[n+Floor[n/3]*Mod[n, 3], {n, 78}]
LinearRecurrence[{0,0,2,0,0,-1},{1,2,3,5,7,6},80] (* Harvey P. Dale, Aug 06 2021 *)

Vec(x*(3*x^4+3*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, May 11 2015

a(n) = n+floor(n/3)*(n mod 3), n = 1, 2, ...
a(n) = 2*a(n-3)-a(n-6). - Colin Barker, May 11 2015
G.f.: x*(3*x^4+3*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, May 11 2015
E.g.f.: (-5+12*x)*exp(x)/9 + (3+2*x)*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)/9 + 5*exp(-x/2)*cos(sqrt(3)*x/2)/9. - Robert Israel, May 11 2015

A142150 The nonnegative integers interleaved with 0's.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 0, 19, 0, 20, 0, 21, 0, 22, 0, 23, 0, 24, 0, 25, 0, 26, 0, 27, 0, 28, 0, 29, 0, 30, 0, 31, 0, 32, 0, 33, 0, 34, 0, 35, 0, 36, 0, 37, 0, 38, 0, 39, 0, 40, 0, 41, 0, 42, 0, 43, 0

Offset: 0

Reinhard Zumkeller, Jul 15 2008

Number of vertical pairs in a wheel with n equal sections. - Wesley Ivan Hurt, Jan 22 2012
Number of even terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013
Also the result of writing n-1 in base 2 and multiplying the last digit with the number with its last digit removed. See A115273 and A257844-A257850 for generalization to other bases. - M. F. Hasler, May 10 2015
Also follows the rule: a(n+1) is the number of terms that are identical with a(n) for a(0..n-1). - Marc Morgenegg, Jul 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Logical Convolutions
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000004, A000007, A000027, A000217, A000326, A001057, A001477, A003817, A008805, A027656, A086099, A142149, A142151, A162610, A191967, A195034, A209297.

a142150 = uncurry (*) . (`divMod` 2) . (+ 1)
a142150_list = scanl (+) 0 a001057_list
-- Reinhard Zumkeller, Apr 02 2012

[n*(1+(-1)^n)/4 : n in [0..100]]; // Wesley Ivan Hurt, Aug 21 2014

&cat[[n, 0]: n in [0..50]]; // Vincenzo Librandi, Oct 31 2016

A142150:=n->n*(1+(-1)^n)/4: seq(A142150(n), n=0..100); # Wesley Ivan Hurt, Aug 21 2014

Table[Mod[Floor[n^2/2], n], {n, 200}] (* Enrique Pérez Herrero, Jul 29 2009 *)
Riffle[Range[0, 50], 0] (* Paolo Xausa, Feb 08 2024 *)

a(n)=!bittest(n,0)*n>>1 \\ M. F. Hasler, May 10 2015

def A142150(n): return (n+1>>1)*(n&1^1) # Chai Wah Wu, Jan 19 2023

a(n) = XOR{k AND (n-k): 0<=k<=n}.
a(n) = (n/2)*0^(n mod 2); a(2*n)=n and a(2*n+1)=0.
a(n) = floor(n^2/2) mod n. - Enrique Pérez Herrero, Jul 29 2009
a(n) = A027656(n-2). - Reinhard Zumkeller, Nov 05 2009
a(n) = Sum_{k=0..n} (k mod 2)*((n-k) mod 2). - Reinhard Zumkeller, Nov 05 2009
a(n+1) = A000217(n) mod A000027(n+1) = A000217(n) mod A001477(n+1). - Edgar Almeida Ribeiro (edgar.a.ribeiro(AT)gmail.com), May 19 2010
From Bruno Berselli, Oct 19 2010: (Start)
a(n) = n*(1+(-1)^n)/4.
G.f.: x^2/(1-x^2)^2.
a(n) = 2*a(n-2)-a(n-4) for n > 3.
Sum_{i=0..n} a(i) = (2*n*(n+1)+(2*n+1)*(-1)^n-1)/16 (see A008805). (End)
a(n) = -a(-n) = A195034(n-1)-A195034(-n-1). - Bruno Berselli, Oct 12 2011
a(n) = A000326(n) - A191967(n). - Reinhard Zumkeller, Jul 07 2012
a(n) = Sum_{i=1..n} floor((2*i-n)/2). - Wesley Ivan Hurt, Aug 21 2014
a(n-1) = floor(n/2)*(n mod 2), where (n mod 2) is the parity of n, or remainder of division by 2. - M. F. Hasler, May 10 2015
a(n) = A158416(n) - 1. - Filip Zaludek, Oct 30 2016
E.g.f.: x*sinh(x)/2. - Ilya Gutkovskiy, Oct 30 2016
a(n) = A000007(a(n-1)) + a(n-2) for n > 1. - Nicolas Bělohoubek, Oct 06 2024

A257850 a(n) = floor(n/10) * (n mod 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 10, multiply the last digit by the number with its last digit removed.
See A142150(n-1) for the base 2 analog and A257843 - A257849 for the base 3 - base 9 variants.
The first 100 terms coincide with those of A035930 (maximal product of any two numbers whose concatenation is n), A171765 (product of digits of n, or 0 for n<10), A257297 ((initial digit of n)*(n with initial digit removed)), but the sequence is of course different from each of these three.
The terms a(10) - a(100) also coincide with those of A007954 (product of decimal digits of n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257849.

Cf. also A007954, A035930, A171765, A257297.

[Floor(n/10)*(n mod 10): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/10] Mod[n, 10], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
a(n,b=10)=(n=divrem(n,b))[1]*n[2]
```

def A257850(n): return n//10*(n%10) # M. F. Hasler, Sep 01 2021

a(n) = 2*a(n-10)-a(n-20). - Colin Barker, May 11 2015

G.f.: x^11*(9*x^8+8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^4-x^3+x^2-x+1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A257844 a(n) = floor(n/4) * (n mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 0, 2, 4, 6, 0, 3, 6, 9, 0, 4, 8, 12, 0, 5, 10, 15, 0, 6, 12, 18, 0, 7, 14, 21, 0, 8, 16, 24, 0, 9, 18, 27, 0, 10, 20, 30, 0, 11, 22, 33, 0, 12, 24, 36, 0, 13, 26, 39, 0, 14, 28, 42, 0, 15, 30, 45, 0, 16, 32, 48, 0, 17, 34, 51, 0, 18, 36

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 4, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A142150 (the base-2 analog), A115273, A257845 - A257850.

[Floor(n/4)*(n mod 4) : n in [0..100]]; // Wesley Ivan Hurt, Jun 22 2015

I:=[0,0,0,0,0,1,2,3]; [n le 8 select I[n] else 2*Self(n-4)-Self(n-8): n in [1..100]]; // Vincenzo Librandi, Jun 23 2015

A257844:=n->floor(n/4)*(n mod 4): seq(A257844(n), n=0..100); # Wesley Ivan Hurt, Jun 22 2015

Table[Floor[n/4] Mod[n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 22 2015 *)

PARI
```
a(n,b=4)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0], Vec(x^5*(3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

def A257844(n): return (n>>2)*(n&3) # Chai Wah Wu, Jan 27 2023

a(n) = 2*a(n-4) - a(n-8), n > 8. - Colin Barker, May 11 2015
G.f.: x^5*(3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, May 11 2015
a(n) = (3 - 2*(-1)^((2*n - 1 + (-1)^n)/4) - (-1)^n)*(2*n - 3 + 2*(-1)^((2*n - 1 + (-1)^n)/4) + (-1)^n)/16. - Wesley Ivan Hurt, Jun 22 2015

A257845 a(n) = floor(n/5) * (n mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 0, 6, 12, 18, 24, 0, 7, 14, 21, 28, 0, 8, 16, 24, 32, 0, 9, 18, 27, 36, 0, 10, 20, 30, 40, 0, 11, 22, 33, 44, 0, 12, 24, 36, 48, 0, 13, 26, 39, 52, 0, 14, 28, 42, 56

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 5, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,0,0,0,0,0,1,2,3,4},80] (* Harvey P. Dale, Aug 15 2021 *)

PARI
```
a(n,b=5)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0,0], Vec(x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-5)-a(n-10). - Colin Barker, May 11 2015

G.f.: x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A257849 a(n) = floor(n/9) * (n mod 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 2, 4, 6, 8, 10, 12, 14, 16, 0, 3, 6, 9, 12, 15, 18, 21, 24, 0, 4, 8, 12, 16, 20, 24, 28, 32, 0, 5, 10, 15, 20, 25, 30, 35, 40, 0, 6, 12, 18, 24, 30, 36, 42, 48, 0, 7, 14, 21, 28, 35, 42, 49, 56, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 9, multiply the last digit by the number with its last digit removed.

See A142150(n-1) for the base 2 analog, and A115273, A257844 - A257850 for the base 3 - base 10 variants.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

[Floor(n/9)*(n mod 9): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/9] Mod[n, 9], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
A257849(n)=n\9*(n%9)
```

concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

from math import prod
def A257849(n): return prod(divmod(n,9)) # Chai Wah Wu, Jan 19 2023

[floor(n/9)*(n % 9)  for n in (0..80)]; # Bruno Berselli, May 11 2015

a(n) = 2*a(n-9)-a(n-18). - Colin Barker, May 11 2015

G.f.: x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2). - Colin Barker, May 11 2015

A257846 a(n) = floor(n/6) * (n mod 6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 2, 4, 6, 8, 10, 0, 3, 6, 9, 12, 15, 0, 4, 8, 12, 16, 20, 0, 5, 10, 15, 20, 25, 0, 6, 12, 18, 24, 30, 0, 7, 14, 21, 28, 35, 0, 8, 16, 24, 32, 40, 0, 9, 18, 27, 36, 45, 0, 10, 20, 30, 40, 50, 0, 11, 22, 33, 44, 55, 0, 12, 24

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 6, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

Table[Floor[n/6]*Mod[n, 6], {n, 120}] (* Michael De Vlieger, May 11 2015 *)

PARI
```
a(n,b=6)=(n=divrem(n,b))[1]*n[2]
```

concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7*(5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-6)-a(n-12). - Colin Barker, May 11 2015

G.f.: x^7*(5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2). - Colin Barker, May 11 2015

A257847 a(n) = floor(n/7) * (n mod 7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 0, 2, 4, 6, 8, 10, 12, 0, 3, 6, 9, 12, 15, 18, 0, 4, 8, 12, 16, 20, 24, 0, 5, 10, 15, 20, 25, 30, 0, 6, 12, 18, 24, 30, 36, 0, 7, 14, 21, 28, 35, 42, 0, 8, 16, 24, 32, 40, 48, 0, 9, 18, 27, 36, 45, 54, 0, 10, 20

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 7, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).

Cf. A194757.

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

Table[Floor[n/7]Mod[n,7],{n,0,80}] (* Harvey P. Dale, Nov 12 2022 *)

PARI
```
a(n,b=7)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0,0,0,0], Vec(x^8*(6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-7)-a(n-14). - Colin Barker, May 11 2015

G.f.: x^8*(6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A257848 a(n) = floor(n/8) * (n mod 8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 6, 8, 10, 12, 14, 0, 3, 6, 9, 12, 15, 18, 21, 0, 4, 8, 12, 16, 20, 24, 28, 0, 5, 10, 15, 20, 25, 30, 35, 0, 6, 12, 18, 24, 30, 36, 42, 0, 7, 14, 21, 28, 35, 42, 49, 0, 8, 16, 24, 32, 40, 48, 56, 0, 9

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 8, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

Table[Floor[n/8]Mod[n,8],{n,0,90}] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7},90] (* Harvey P. Dale, Nov 05 2023 *)

PARI
```
a(n,b=8)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0,0,0,0,0], Vec(x^9*(7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

def A257848(n): return (n>>3)*(n&7) # Chai Wah Wu, Jan 19 2023

a(n) = 2*a(n-8)-a(n-16). - Colin Barker, May 11 2015

G.f.: x^9*(7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)^2). - Colin Barker, May 11 2015

cp's OEIS Frontend

A115274 a(n) = n + A115273(n), where A115273(n) = 0 for n = 1..3.

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A142150 The nonnegative integers interleaved with 0's.

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A257850 a(n) = floor(n/10) * (n mod 10).

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A257844 a(n) = floor(n/4) * (n mod 4).

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A257845 a(n) = floor(n/5) * (n mod 5).

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A257849 a(n) = floor(n/9) * (n mod 9).

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