A011858 -id:A011858 - cp's OEIS frontend

A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...

0, 4, 6, 18, 22, 42, 48, 76, 84, 120, 130, 174, 186, 238, 252, 312, 328, 396, 414, 490, 510, 594, 616, 708, 732, 832, 858, 966, 994, 1110, 1140, 1264, 1296, 1428, 1462, 1602, 1638, 1786, 1824, 1980, 2020, 2184, 2226, 2398, 2442, 2622, 2668, 2856, 2904, 3100

Offset: 1

Bruno Berselli, Nov 14 2012

Equivalently, numbers m such that 20*m+1 is a square.
Also, integer values of h*(h+1)/5.
More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:
. generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);
. n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;
. first differences: (n-1)*((-1)^n*(k-2)+k)/2;
. b(2n+1)-b(2n) = 2*n (independent from k);
. (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.

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

Subsequence of A011858.
Cf. A090771: square roots of 20*a(n)+1 (see the first comment).
Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.
Cf. similar sequences listed in A219257.

k:=5; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

I:=[0,4,6,18,22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (2 + x + 2 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,6,18,22},50] (* Harvey P. Dale, Jan 21 2015 *)

G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.
a(n) = 2*A057569(n) = A008851(n+1)*A047208(n)/5.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2015
Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - Amiram Eldar, Mar 15 2022
a(n) = A132356(n-1)/2, n >= 1. - Bernard Schott, Mar 15 2022

A008738 a(n) = floor((n^2 + 1)/5).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 10, 13, 16, 20, 24, 29, 34, 39, 45, 51, 58, 65, 72, 80, 88, 97, 106, 115, 125, 135, 146, 157, 168, 180, 192, 205, 218, 231, 245, 259, 274, 289, 304, 320, 336, 353, 370, 387, 405, 423, 442, 461, 480, 500, 520, 541, 562, 583, 605, 627, 650, 673

Offset: 0

N. J. A. Sloane

Without initial zeros, Molien series for 3-dimensional group [2+,n] = 2*(n/2).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A011858. Partial sums of A288156.

List([0..60], n-> Int((n^2 + 1)/5)); # G. C. Greubel, Aug 03 2019

[(n^2+1) div 5: n in [0..60]]; // Bruno Berselli, Oct 28 2011

Floor[(Range[0,60]^2 + 1)/5] (* G. C. Greubel, Aug 03 2019 *)

PARI
```
a(n)=(n^2+1)\5;
```

[floor((n^2+1)/5) for n in (0..60)] # G. C. Greubel, Aug 03 2019

G.f.: x^2*(1+x^3)/((1-x)^2*(1-x^5)) = x^2*(1+x)*(1-x+x^2)/( (1-x)^3 *(1+x+x^2+x^3+x^4) ).

a(n+2)= A249020(n) + A249020(n-1). - R. J. Mathar, Aug 11 2021

More terms from Philip Mummert (s1280900(AT)cedarville.edu), May 10 2000

A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 19 2011

nonn

The defining [sum] is equivalent to
...
a(n)=[n(n+1)r/2]-sum{[k*r] : 1<=k<=n},
...
where []=floor and r=sqrt(2).  Let s(n) denote the n-th partial sum of the sequence a; then the difference sequence d defined by d(n)=s(n+1)-s(n) gives the runlengths of a.
...
Examples:
...
r...........a........s....
1/2......A002265...A001972
1/3......A002264...A001840
2/3......A002264...A001840
1/4......A194220...A194221
1/5......A194222...A118015
2/5......A057354...A011858
3/5......A194222...A118015
4/5......A057354...A011858
1/6......A194223...A194224
3/7......A057357...A194229
1/8......A194235...A194236
3/8......A194237...A194238
sqrt(2)..A194161...A194162
sqrt(3)..A194163...A194164
sqrt(5)..A194169...A194170
sqrt(6)..A194173...A194174
tau......A194165...A194166; tau=(1+sqrt(5))/2
e........A194200...A194201
2e.......A194202...A194203
e/2......A194204...A194205
pi.......A194206...A194207

			a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]
    =[.718+.436+.154+.873+.591]
    =[2.77423]=2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A194201.

r = E;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]  (* A194200 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A194201 *)

A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.

Original entry on oeis.org

1, 1, 5, 9, 17, 25, 33, 45, 57, 73, 89, 105, 125, 145, 169, 193, 217, 245, 273, 305, 337, 369, 405, 441, 481, 521, 561, 605, 649, 697, 745, 793, 845, 897, 953, 1009, 1065, 1125, 1185, 1249, 1313, 1377, 1445, 1513, 1585, 1657, 1729, 1805, 1881

Offset: 0

Clark Kimberling, Apr 11 2012

For a guide to related sequences, see A211422.

Also, a(n) is the number of ordered pairs (w,x) with both terms in {-n,...,0,...,n} and w+2x divisible by 5. If (w,x) is such a pair it is easy to see that (-x,w), (-w,-x), and (x,-w) also are such pairs. If both w and x are nonzero these four pairs lie one in each quadrant. If one of w or x is zero, the other must be a multiple of 5. This means that a(n) equals 4*A211523(n) (the nonzero pairs) plus 4*floor(n/5) + 1 (pairs with w or x equal to zero). Since the sequences A211523(n), floor(n/5), and the constant sequence all satisfy the recurrence conjectured in the formula section, a(n) must also satisfy the recurrence, so this proves the conjecture. - Pontus von Brömssen, Jan 17 2020

Pontus von Brömssen, Table of n, a(n) for n = 0..1024
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A211422, A211523.

a:=[]; for n in [0..50] do m:=0; for i, j in [-n..n] do if (i+2*j) mod 5 eq 0  then m:=m+1; end if; end for; Append(~a, m); end for; a; // Marius A. Burtea, Jan 19 2020

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)))); // Marius A. Burtea, Jan 19 2020

t[n_] := t[n] = Flatten[Table[w + 2 x + 5 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211434 *)
(t - 1)/4                    (* A011858 *)

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.
(End)
a(n) = (4*n*(n+1) + c(n))/5, where c(n) is 5 if n is 0 or 4 (mod 5), -3 if n is 1 or 3 (mod 5), and 1 if n is 2 (mod 5). - Pontus von Brömssen, Jan 17 2020

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 49, 57, 66, 75, 85, 95, 106, 118, 130, 143, 156, 170, 185, 200, 216, 232, 249, 267, 285, 304, 323, 343, 364, 385, 407, 429, 452, 476, 500, 525, 550, 576, 603, 630, 658, 686, 715, 745, 775, 806, 837, 869, 902, 935

Offset: 1

Clark Kimberling, Jul 08 2013

See A227347.

			a(1) = floor(3/5) = 0; a(2) = floor(6/5) = 1; a(3) = a(2) + floor(9/5) = 2; a(4) = a(3) + floor(12/5) = 4.

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

Cf. A227347, A130520, A011858, A033437.

Cf. A057355 (first differences).

z = 150; r = 3/5; k = 1; a[n_] := Sum[Floor[r*x^k], {x, 1, n}]; t = Table[a[n], {n, 1, z}]

a(n) = (3*n^2-n)\10; \\ Kevin Ryde, Mar 15 2022

a = lambda n: n*(3*n-1)//10 # Gennady Eremin, Mar 20 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
G.f.: (x*(1 + x^2 + x^3))/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
According to Wolfram Alpha, a(n) = floor(Re(E(n^2|Pi))) where E(x|m) is the incomplete elliptic integral of the second kind. - Kritsada Moomuang, Jan 28 2022
a(n) = a(n-1) + floor(3*n/5), n > 1. - Gennady Eremin, Mar 15 2022
a(n) = floor(n*(3*n-1)/10). - Kevin Ryde, Mar 15 2022

A376080 a(n) is the highest degree of the rational function in the recursive substitution {y1, y2} -> {y2, (y2 + 1)/(y1*y2)} after n steps.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 10, 13, 16, 20, 25, 29, 34, 40, 46, 52, 59, 67, 74, 82, 91, 100, 109, 119, 130, 140, 151, 163, 175, 187, 200, 214, 227, 241, 256, 271, 286, 302, 319, 335, 352, 370, 388, 406, 425, 445, 464, 484, 505, 526, 547, 569, 592, 614, 637, 661, 685, 709, 734, 760, 785, 811, 838, 865

Offset: 0

Thomas Scheuerle, Sep 09 2024

An example where the degree of the n-th iterate of a rational map exhibits polynomial growth. Also an example for exponential growth was given in the thesis from Khaled Hamad by A011782.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Khaled Hamad, Laurentification, Thesis (2017). La Trobe University.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A011858, A058937.

Cf. A011782 (highest degree of the rational function in the substitution: {y1, y2} -> {y2, y2 + y1/y2}).

A376080[n_] := Ceiling[(3*n*(n - 1) + 8)/14];
Array[A376080, 100, 0] (* Paolo Xausa, Sep 23 2024 *)

r(v) = [v[2], (v[2]+1)/(v[1]*v[2])];
a(n) = {my(v = [x,x]); if(n < 2, 1,for(k=0, n-2, v = r(v)); poldegree(numerator(v[2])))};

G.f.: (1 - x + x^3 + x^4 - x^5 + x^6 + x^8)/(1 - 2*x + x^2 - x^7 + 2*x^8 - x^9).
a(n) = ceiling((3*n^2 - 3*n + 8)/14).
a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9).
a(n) = a(n-7) + 3*(n-7) + 9.
(2*a(n+6) - a(n+5) - 2*a(n-1) + a(n-2) - 9)/3  = n.

cp's OEIS Frontend

A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...

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A008738 a(n) = floor((n^2 + 1)/5).

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A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

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A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.

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A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

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A376080 a(n) is the highest degree of the rational function in the recursive substitution {y1, y2} -> {y2, (y2 + 1)/(y1*y2)} after n steps.

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A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...