A047521 -id:A047521 - cp's OEIS frontend

A274406 Numbers m such that 9 divides m*(m + 1).

0, 8, 9, 17, 18, 26, 27, 35, 36, 44, 45, 53, 54, 62, 63, 71, 72, 80, 81, 89, 90, 98, 99, 107, 108, 116, 117, 125, 126, 134, 135, 143, 144, 152, 153, 161, 162, 170, 171, 179, 180, 188, 189, 197, 198, 206, 207, 215, 216, 224, 225, 233, 234, 242, 243, 251, 252, 260, 261, 269

Offset: 1

Bruno Berselli, Jun 20 2016

Equivalently, numbers congruent to 0 or 8 mod 9.

Terms of A007494 with indices in A047264. Also, terms of A060464 with indices in A047335.

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

Cf. A008591 (first bisection), A010689 (first differences), A017257 (second bisection).
Cf. similar sequences in which m*(m+1) is divisible by k: A014601 (k=4), A047208 (k=5), A007494 (k=3 and 6), A047335 (k=7), A047521 (k=8), this sequence (k=9).
Cf. A301451: numbers congruent to {1, 7} mod 9; A193910: numbers congruent to {2, 6} mod 9.

[n: n in [0..300] | IsDivisibleBy(n*(n+1),9)];

Select[Range[0, 300], Divisible[# (# + 1), 9] &]

for(n=0, 300, if(n*(n+1)%9==0, print1(n", ")))

[n for n in range(300) if 9.divides(n*(n+1))]

G.f.: x^2*(8 + x)/((1 + x)*(1 - x)^2).
a(n) = (18*n + 7*(-1)^n - 11)/4. Therefore: a(2*m) = 9*m-1, a(2*m+1) = 9*m. It follows that a(j)+a(k) and a(j)*a(k) belong to the sequence if j and k are not both even.
a(n) = -A090570(-n+2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(2*r+1) + a(2*r+s+1) = a(4*r+s+1) and a(2*r) + a(2*r+2*s+1) = a(4*r+2*s). A particular case provided by these identities: a(n) = a(n - 2*floor(n/6)) + a(2*floor(n/6) + 1).
E.g.f.: 1 + ((9*x - 2)*cosh(x) + 9*(x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2021

A176415 Periodic sequence: repeat 7,1.

Original entry on oeis.org

7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7

Offset: 0

Klaus Brockhaus, Apr 17 2010

Interleaving of A010727 and A000012.
Also continued fraction expansion of (7+sqrt(77))/2.
Also decimal expansion of 71/99.
Essentially first differences of A047521.
Binomial transform of A176414.
Inverse binomial transform of 2*A020707 preceded by 7.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 4*x^3 + 10*x^4 + 10*x^5 + ... is the o.g.f. for A058187. - Peter Bala, Mar 13 2015

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

Cf. A010727 (all 7's sequence), A000012 (all 1's sequence), A092290 (decimal expansion of (7+sqrt(77))/2), A010688 (repeat 1, 7), A047521 (congruent to 0 or 7 mod 8), A176414 (expansion of (7+8*x)/(1+2*x)), A020707 (2^(n+2)), A058187.

&cat[ [7, 1]: n in [0..52] ];
[ 4+3*(-1)^n: n in [0..104] ];

PadRight[{},120,{7,1}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=7-n%2*6 \\ Charles R Greathouse IV, Oct 28 2011

a(n) = 4+3*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 7, a(1) = 1.
a(n) = -a(n-1)+8 for n > 0; a(0) = 7.
a(n) = 7*((n+1) mod 2)+(n mod 2).
a(n) = A010688(n+1).
G.f.: (7+x)/(1-x^2).
Dirichglet g.f.: (1+6*2^(-s))*zeta(s). - R. J. Mathar, Apr 06 2011
Multiplicative with a(2^e) = 7, and a(p^e) = 1 for p >= 3. - Amiram Eldar, Jan 01 2023

A165719 Integers of the form k*(k+9)/8.

Original entry on oeis.org

14, 17, 45, 50, 92, 99, 155, 164, 234, 245, 329, 342, 440, 455, 567, 584, 710, 729, 869, 890, 1044, 1067, 1235, 1260, 1442, 1469, 1665, 1694, 1904, 1935, 2159, 2192, 2430, 2465, 2717, 2754, 3020, 3059, 3339, 3380, 3674, 3717, 4025, 4070, 4392, 4439, 4775

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Only one term is a prime number (17). Are all others composite?
There is no prime other than 17 in the first 1 million terms. - Harvey P. Dale, Jan 07 2020
Integers of the form k+k*(k+1)/8 = k+A000217(k)/4; for k see A047521, for A000217(k)/4 see A154260.

			for k = 1,2,..., k(k+9)/8 is 5/4, 11/4, 9/2, 13/2, 35/4, 45/4, 14, 17,.. and the integer values out of these become the sequence.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A047521, A154260, A165717, A165718.

q=4;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst
Select[Table[(n(n+9))/8,{n,200}],IntegerQ] (* or *) Rest[Flatten[Table[ {9n+8n^2,14+23n+8n^2},{n,0,30}]]] (* or *) LinearRecurrence[{1,2,-2,-1,1},{14,17,45,50,92},60] (* Harvey P. Dale, Jan 07 2020 *)

From R. J. Mathar, Sep 25 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = 2*n^2 + 6*n + 9/4 - 3*(-1)^n*(2*n+3)/4.
G.f.: x*(-14-3*x+x^3)/((1+x)^2 * (x-1)^3 ). (End)
Sum_{n>=1} 1/a(n) = 89/81 - (sqrt(2)+1)*Pi/9. - Amiram Eldar, Jul 26 2024

Definition simplified by R. J. Mathar, Sep 25 2009

A047589 Numbers that are congruent to {6, 7} mod 8.

Original entry on oeis.org

6, 7, 14, 15, 22, 23, 30, 31, 38, 39, 46, 47, 54, 55, 62, 63, 70, 71, 78, 79, 86, 87, 94, 95, 102, 103, 110, 111, 118, 119, 126, 127, 134, 135, 142, 143, 150, 151, 158, 159, 166, 167, 174, 175, 182, 183, 190, 191, 198, 199, 206, 207, 214, 215, 222, 223, 230, 231

Offset: 1

N. J. A. Sloane

These are the values of n for which binomial(n,6) is odd. See Maple code. - Gary Detlefs, Nov 29 2011

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

Union of A017137 and A004771.

Cf. A047451, A047521.

for i from 1 to 240 do if(floor((i mod 8)/6) <>0) then print(i) fi od; # Gary Detlefs, Nov 30 2011

LinearRecurrence[{1,1,-1},{6,7,14},60] (* Harvey P. Dale, Sep 11 2017 *)

a(n) = 8*n-a(n-1)-3 with n>1, a(1)=6. - Vincenzo Librandi, Aug 06 2010
a(n) = 6*floor((n-1)/2) + n + 5. - Gary Detlefs, Nov 29 2011
a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: x*(6+x+x^2)/((1-x)^2*(1+x)). - Colin Barker, Mar 18 2012
a(n) = (1-3*(-1)^n+8*n)/2. - Colin Barker, May 14 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 - log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A317806 Number of set partitions of [k] into 4 blocks with equal element sum, where k is the n-th positive integer that allows such a partition.

Original entry on oeis.org

1, 1, 871, 2650, 9462094, 31650271, 171019406993, 595828948333, 4107584704538352, 14702365152800667, 118513210888679225825, 432046935173440593804, 3881432331405193485285518, 14337098117309087488187476, 139477762791757859249400365738, 520312171172086830267314753894

Offset: 1

Alois P. Heinz, Aug 07 2018

nonn

k = 7, 8, 15, 16, 23, ... A047521(n+1) for n = 1, 2, 3, 4, 5, ... .

			a(1) = 1: 16|25|34|7 with k = 7.
a(2) = 1: 18|27|36|45 with k = 8.

Wikipedia, Partition of a set

Cf. A047521, A275714.

b:= proc() option remember; local i, j, t; `if`(args[1]=0,
      `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(
      `if`(args[j] -args[nargs]<0, 0, b(sort([seq(args[i]-
      `if`(i=j, args[nargs], 0), i=1..nargs-1)])[],
                args[nargs]-1)), j=1..nargs-1))
    end:
a:= proc(n) option remember; (k-> (m->
      b((m/4)$4, k)/24)(k*(k+1)/2))(4*n+3/2*(1-(-1)^n))
    end:
seq(a(n), n=1..8);

a(n) = A275714(A047521(n+1),4).

A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

Original entry on oeis.org

0, 2, 7, 15, 25, 37, 52, 70, 90, 112, 137, 165, 195, 227, 262, 300, 340, 382, 427, 475, 525, 577, 632, 690, 750, 812, 877, 945, 1015, 1087, 1162, 1240, 1320, 1402, 1487, 1575, 1665, 1757, 1852, 1950, 2050, 2152, 2257

Offset: 0

Paul Curtz, Feb 14 2020

a(-2)=2, a(-1)=0. 4 evens followed by 4 odds.
Last digit is only 0, 2, 5, 7.
The vertical spoke S-N of the pentagonal spiral for A004526.
                          37
                      37  25  25
                  36  24  15  15  26
              36  24  14   7   8  16  26
          35  23  14   7   2   3   8  16  27
      35  23  13   6   2   0   0   3   9  17  27
        34  22  13   6   1   1   4   9  17  28
          34  22  12   5   5   4  10  18  28
            33  21  12  11  11  10  18  29
              33  21  20  20  19  19  29
                32  32  31  31  30  30
Rank of multiples of 10: 0, 7, 8, 15, 16, ... = A047521. Compare to A154260 in the formula.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A004526, A033429, A062786, A168668, A135706, A147874, 2*A147875 (all in the spiral).

Cf. A005408, A047521, A154260

CoefficientList[Series[x (2 + x + 2 x^2)/((1 - x)^3*(1 + x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Feb 14 2020 *)

concat(0, Vec(x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ Colin Barker, Feb 14 2020

a(-1-n) = a(n).
a(2*n) + a(1+2*n) = 2, 22, 62, ... = A273366(n).
Second differences give the sequence of period 4: repeat [3, 3, 2, 2].
From Colin Barker, Feb 14 2020: (Start)
G.f.: x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
Multiples of 10: 10*(0, 7, 9, 30, 34, ... = A154260).
4*a(n) = A087960(n) +5*n -1 +5*n^2. - R. J. Mathar, Feb 28 2020

A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Sep 14 2022

This sequence is directly related to A355278.
The function f is a bijection from the nonnegative integers to the integers (Z).
The nonnegative integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the additive group Z (f and g act as isomorphisms).
As a consequence, each row and each column is a permutation of the nonnegative integers.

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12
    1 |  1   3   0   5   2   7   4   9   6  11   8  13  10
    2 |  2   0   4   1   6   3   8   5  10   7  12   9  14
    3 |  3   5   1   7   0   9   2  11   4  13   6  15   8
    4 |  4   2   6   0   8   1  10   3  12   5  14   7  16
    5 |  5   7   3   9   1  11   0  13   2  15   4  17   6
    6 |  6   4   8   2  10   0  12   1  14   3  16   5  18
    7 |  7   9   5  11   3  13   1  15   0  17   2  19   4
    8 |  8   6  10   4  12   2  14   0  16   1  18   3  20
    9 |  9  11   7  13   5  15   3  17   1  19   0  21   2
   10 | 10   8  12   6  14   4  16   2  18   0  20   1  22
   11 | 11  13   9  15   7  17   5  19   3  21   1  23   0
   12 | 12  10  14   8  16   6  18   4  20   2  22   0  24

Cf. A001057, A014601, A014681, A047264, A047521, A355278, A357144.

f(n) = - (-1)^n * ((n+1)\2)
g(n) = if (n<=0, -2*n, 2*n-1)
A(n, k) = g(f(n) + f(k))

A355278(n+1, k+1) = prime(1 + A(n, k)) (where prime(m) denotes the m-th prime number).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, A014681(n)) = 0.
A(m, A(n, k)) = A(A(m, n), k).
A(n, n) = A014601(n).
A(n, A(n, n)) = A047264(n+1).
A(A(n, n), A(n, n)) = A047521(n+1).

cp's OEIS Frontend

A274406 Numbers m such that 9 divides m*(m + 1).

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A176415 Periodic sequence: repeat 7,1.

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A165719 Integers of the form k*(k+9)/8.

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A047589 Numbers that are congruent to {6, 7} mod 8.

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A317806 Number of set partitions of [k] into 4 blocks with equal element sum, where k is the n-th positive integer that allows such a partition.

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A332495 a(n-2) = a(n-6) + 5*(1+2*n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

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A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

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A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.