A056865 -id:A056865 - cp's OEIS frontend

A056864 Nearest integer to n^2/10.

0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 26, 29, 32, 36, 40, 44, 48, 53, 58, 63, 68, 73, 78, 84, 90, 96, 102, 109, 116, 123, 130, 137, 144, 152, 160, 168, 176, 185, 194, 203, 212, 221, 230, 240, 250, 260, 270, 281, 292, 303, 314, 325

Offset: 0

N. J. A. Sloane, Sep 02 2000

nonn

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

Cf. A056865.

A056864 := proc(n)
    floor(n^2/10+1/2) ;
end proc:
seq(A056864(n),n=0..100) ; # R. J. Mathar, Mar 08 2016

Table[Floor[n^2/10+1/2],{n,0,100}] (* Harvey P. Dale, Nov 01 2020 *)

a(n) = round(n^2/10); \\ Michel Marcus, Mar 08 2016

G.f.: -x^3*(1+x^2)*(x^4-x^2+1) / ( (1+x) *(x^4-x^3+x^2-x+1) *(x^4+x^3+x^2+x+1) *(x-1)^3 ). - R. J. Mathar, Mar 08 2016

A134119 a(n) = floor(n^2/10) - floor((n-1)^2/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17

Offset: 0

Alexander R. Povolotsky, Jan 12 2008

Note that for n >=1 there is a pattern that keeps steadily alternating between 4 terms and 6 terms for the each two consecutive groups. The terms value remains the same within each 4-term or 6-term group, while during the switch from the 4-group to the 6-group and then back from the 6-group to the 4-group, etc., the term value is getting bumped by 1.

Assuming this obeys the recurrence a(n) = a(n-10) + 2, this has generating function G(x) = x^4*(1+x^4)/[(-1+x)^2*(x+1)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)] = (1 - 3x^2 - 3x^3)/[10(x^4 + x^3 + x^2 + x + 1)]+1/[10(x+1)] + 1/[5(-1+x)^2] +(-1 + 2x - 3x^2 - x^3)/[10(x^4 - x^3 + x^2 - x + 1)] + 3/[10(-1+x)]. The first term can be rewritten as a linear superposition of A104384(n), A104384(n+2), A103483(n+3); the second, ~1/(x+1), with the alternating A033999, the third component ~1/(x-1)^2 with a(n)=n+1, the next ~1/(x^4 - x^3 + x^2 - x + 1) = A014019 and the last is proportional to 1/(1-x) = A000012. So a(n) is a sum of these sequences. - R. J. Mathar, Jan 16 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A000012, A014019, A033999, A056865.

Table[Floor[n^2/10] - Floor[(n - 1)^2/10], {n, 0, 50}] (* G. C. Greubel, Feb 22 2017 *)

PARI
```
a(n)= floor(n^2/10) - floor((n-1)^2/10)
```

Empirical g.f.: x^4*(x^4+1) / (x^11 - x^10 - x + 1). - Colin Barker, Aug 08 2013

The above conjectured g.f. is correct. - Sela Fried, Dec 08 2024

More terms from N. J. A. Sloane, Jan 22 2008

A342112 Drop the final digit of n^5.

Original entry on oeis.org

0, 0, 3, 24, 102, 312, 777, 1680, 3276, 5904, 10000, 16105, 24883, 37129, 53782, 75937, 104857, 141985, 188956, 247609, 320000, 408410, 515363, 643634, 796262, 976562, 1188137, 1434890, 1721036, 2051114, 2430000, 2862915, 3355443, 3913539, 4543542, 5252187, 6046617

Offset: 0

Stefano Spezia, Feb 28 2021

Why exponent 5? Because it is the smallest nontrivial exponent e such that for an integer k not ending in 0, 1, 5 and 6, k^e has the same unit digit of k in base 10.

Index entries for linear recurrences with constant coefficients, signature (5,-10, 10,-5,1,0,0,0,0,1,-5,10,-10,5,-1).
Index entries for sequences related to final digits of numbers.

Cf. A000584, A010879, A016813, A056865, A059995, A061167, A164938.

Table[(n^5-Last[IntegerDigits[n]])/10,{n,0,36}]

G.f.: x^2*(3 + 9*x + 12*x^2 + 12*x^3 + 12*x^4 + 12*x^5 + 12*x^6 + 12*x^7 + 13*x^8 + 8*x^9 + 15*x^10 - x^11 + x^12)/((1 - x)^6*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + a(n-10) - 5*a(n-11) + 10*a(n-12) - 10*a(n-13) + 5*a(n-14) - a(n-15) for n > 14.
a(n) = floor(n^5/10).
a(n) = (A000584(n) - A010879(n))/10.
a(n) = A164938(n) + A059995(n).

cp's OEIS Frontend

A056864 Nearest integer to n^2/10.

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A134119 a(n) = floor(n^2/10) - floor((n-1)^2/10).

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A342112 Drop the final digit of n^5.

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