A288156 -id:A288156 - cp's OEIS frontend

A289120 a(n) is the number of odd integers divisible by 7 in ]2(n-1)^2, 2n^2[.

0, 0, 1, 0, 1, 2, 1, 2, 2, 3, 2, 3, 4, 3, 4, 4, 5, 4, 5, 6, 5, 6, 6, 7, 6, 7, 8, 7, 8, 8, 9, 8, 9, 10, 9, 10, 10, 11, 10, 11, 12, 11, 12, 12, 13, 12, 13, 14, 13, 14, 14, 15, 14, 15, 16, 15, 16, 16, 17, 16, 17, 18, 17, 18, 18, 19, 18, 19, 20, 19, 20, 20, 21, 20, 21, 22, 21, 22, 22, 23, 22, 23, 24, 23, 24, 24, 25, 24, 25, 26, 25, 26, 26, 27, 26, 27, 28, 27, 28, 28, 29

Offset: 0

Ralf Steiner, Jun 25 2017

This sequence has the form (0+2k,0+2k,1+2k,0+2k,1+2k,2+2k,1+2k) for k>=0.

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

Cf. A289133, A289122, A288156, A004523.

Table[Count[Mod[Table[2 ((n - 1)^2 + k) - 1, {k, 1, 2 n - 1}], 7],
  0], {n, 0, 100}]

concat(vector(2), Vec(x^2*(1 + x)*(1 - x + x^2)^2 / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^100))) \\ Colin Barker, Jul 02 2017

From Colin Barker, Jul 02 2017: (Start)
G.f.: x^2*(1 + x)*(1 - x + x^2)^2 / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8) for n>7.
(End)

A289122 a(n) is number of odd integers divisible by 11 in the interval ]2(n-1)^2, 2n^2[.

Original entry on oeis.org

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

Offset: 0

Ralf Steiner, Jun 25 2017

This sequence has the form (0+2k,0+2k,0+2k,1+2k,0+2k,1+2k,1+2k,1+2k,2+2k, 1+2k,2+2k) for k>=0.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A289120, A289133, A288156, A004523, A289139.

Table[Count[Mod[Table[2((n-1)^2 +k) -1,{k,1,2n-1}],11],0],{n,0,50}]
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1},{0,0,0,1,0,1,1,1,2,1,2,2},90] (* Harvey P. Dale, Aug 24 2017 *)

a(n) = sum(k=2*(n-1)^2, 2*n^2, ((k % 2) && ((k % 11) == 0))); \\ Michel Marcus, Jun 26 2017

a(n + 11*k) = a(n) + 2*k. - David A. Corneth, Jun 25 2017

G.f.: (x^10-x^9+x^8+x^5-x^4+x^3)/(x^12-x^11-x+1). - Alois P. Heinz, Jun 26 2017

A289133 a(n) is the number of odd integers divisible by 9 in ]2(n-1)^2, 2n^2[.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Ralf Steiner, Jun 25 2017

This sequence has the form (0+2k,0+2k,0+2k,1+2k,1+2k,1+2k,1+2k,1+2k,2+2k) for k>=0.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A289120, A289122, A288156, A004523, A289139.

Table[Count[Mod[Table[2((n-1)^2 +k)-1,{k,1,2 n-1}],9],0],{n,0,50}]

a(n) = sum(k=2*(n-1)^2, 2*n^2, ((k % 2) && ((k % 9) == 0))); \\ Michel Marcus, Jun 26 2017

a(n + 9*k) = a(n) + 2*k.

G.f.: (x^8+x^3)/(x^10-x^9-x+1). - Alois P. Heinz, Jun 26 2017

More terms from Michel Marcus, Jun 26 2017

A289139 a(n) is the number of odd integers divisible by 7 in ]4(n-1)^2, 4n^2[.

Original entry on oeis.org

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

Offset: 0

Ralf Steiner, Jun 26 2017

This sequence has the form (0+4k,0+4k,1+4k,2+4k,2+4k,2+4k,3+4k) for k>=0.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A289120, A289133, A289122, A288156, A004523.

Table[Count[Mod[Table[2(2(n-1)^2 +k)-1, {k,1,4 n-2}],7],0], {n,0,100}]
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,0,1,2,2,2,3,4},80] (* Harvey P. Dale, Aug 10 2025 *)

a(n + 7*k) = a(n) + 4*k.

G.f.: (x^7+x^6+x^3+x^2)/(x^8-x^7-x+1). - Alois P. Heinz, Jun 26 2017

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

A289195 a(n) is the number of odd integers divisible by 5 in ]4(n-1)^2, 4n^2[.

Original entry on oeis.org

0, 0, 2, 2, 2, 4, 4, 6, 6, 6, 8, 8, 10, 10, 10, 12, 12, 14, 14, 14, 16, 16, 18, 18, 18, 20, 20, 22, 22, 22, 24, 24, 26, 26, 26, 28, 28, 30, 30, 30, 32, 32, 34, 34, 34, 36, 36, 38, 38, 38, 40, 40, 42, 42, 42, 44, 44, 46, 46, 46, 48, 48, 50, 50, 50, 52, 52, 54

Offset: 0

Ralf Steiner, Jun 28 2017

This sequence has the form (0+4k,0+4k,2+4k,2+4k,2+4k) for k>=0.

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

Cf. A289199, A289139, A289120, A289133, A289122, A288156, A004523.

Table[Count[Mod[Table[2(2(n-1)^2+k)-1,{k,1,4 n-2}],5],0],{n,0,50}]

concat(vector(2), Vec(2*x^2*(1 + x)*(1 - x + x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Colin Barker, Jul 04 2017

a(n + 5*k) = a(n) + 4*k.
From Colin Barker, Jul 04 2017: (Start)
G.f.: 2*x^2*(1 + x)*(1 - x + x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
(End)

A289199 a(n) is the number of odd integers divisible by 13 in the open interval (12(n-1)^2, 12n^2).

Original entry on oeis.org

0, 0, 2, 2, 3, 5, 5, 6, 7, 7, 9, 10, 10, 12, 12, 14, 14, 15, 17, 17, 18, 19, 19, 21, 22, 22, 24, 24, 26, 26, 27, 29, 29, 30, 31, 31, 33, 34, 34, 36, 36, 38, 38, 39, 41, 41, 42, 43, 43, 45, 46, 46, 48, 48, 50, 50, 51, 53, 53, 54, 55, 55, 57, 58, 58, 60, 60, 62, 62, 63, 65

Offset: 0

Ralf Steiner, Jun 28 2017

This sequence has the form (0+12k, 0+12k, 2+12k, 2+12k, 3+12k, 5+12k, 5+12k, 6+12k, 7+12k, 7+12k, 9+12k, 10+12k, 10+12k) for k >= 0.
Theorems: A) Generally for an interval (2*m*(n-1)^2,2*m*n^2) and a divisor d with 2*m < d there is a unique d-length form (e_i+2*m*k)_{i=0..d-1, k>=0} with e_i in [0,2*m]; here m = 6, d = 13.
B) Sum_{i=0..d-1}e_i = m*(d-2); here 66 = 6*(13-2).
Proof:
A) In d consecutive intervals
(2*m*(n-1)^2,2*m*(n+2)^2) there are m*d*(2*k+d) consecutive odd numbers and therefore m*(2*k+d) multiples of d where k=floor((n-1)/d).
B) With initial value a(0)=0 we have a(d)=2*m and thus Sum_{i=0..d-1} e_i = Sum_{i=1..d}a(i)-a(d) = m(2*0+d)-2*m = m*(d-2). Q.E.D.

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

Cf. A004523, A288156, A289120, A289122, A289133, A289139, A289195.

Table[Count[Mod[Table[2(6(n-1)^2 +k)-1,{k,12 n-6}],13],0],{n,0,70}]

a(n)=(12*n^2+12)\26 - (12*n^2-24*n+25)\26 \\ Charles R Greathouse IV, Jun 29 2017

concat(vector(2), Vec(x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)) + O(x^100))) \\ Colin Barker, Jul 03 2017

a(n + 13*k) = a(n) + 12*k.
a(n) = 12n/13 + O(1). - Charles R Greathouse IV, Jun 29 2017
From Colin Barker, Jul 03 2017: (Start)
G.f.: x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)).
a(n) = a(n-1) + a(n-13) - a(n-14) for n>13.
(End)

cp's OEIS Frontend

A289120 a(n) is the number of odd integers divisible by 7 in ]2*(n-1)^2, 2*n^2[.

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A289122 a(n) is number of odd integers divisible by 11 in the interval ]2*(n-1)^2, 2*n^2[.

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A289133 a(n) is the number of odd integers divisible by 9 in ]2*(n-1)^2, 2*n^2[.

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A289139 a(n) is the number of odd integers divisible by 7 in ]4*(n-1)^2, 4*n^2[.

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

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A289195 a(n) is the number of odd integers divisible by 5 in ]4*(n-1)^2, 4*n^2[.

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A289199 a(n) is the number of odd integers divisible by 13 in the open interval (12*(n-1)^2, 12*n^2).

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A289120 a(n) is the number of odd integers divisible by 7 in ]2(n-1)^2, 2n^2[.

A289122 a(n) is number of odd integers divisible by 11 in the interval ]2(n-1)^2, 2n^2[.

A289133 a(n) is the number of odd integers divisible by 9 in ]2(n-1)^2, 2n^2[.

A289139 a(n) is the number of odd integers divisible by 7 in ]4(n-1)^2, 4n^2[.

A289195 a(n) is the number of odd integers divisible by 5 in ]4(n-1)^2, 4n^2[.

A289199 a(n) is the number of odd integers divisible by 13 in the open interval (12(n-1)^2, 12n^2).