A016885 -id:A016885 - cp's OEIS frontend

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

5, 8, 13, 11, 18, 25, 14, 23, 32, 41, 17, 28, 39, 50, 61, 20, 33, 46, 59, 72, 85, 23, 38, 53, 68, 83, 98, 113, 26, 43, 60, 77, 94, 111, 128, 145, 29, 48, 67, 86, 105, 124, 143, 162, 181, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Vincenzo Librandi, Jan 13 2009

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   5;
   8, 13;
  11, 18, 25;
  14, 23, 32, 41;
  17, 28, 39, 50,  61;
  20, 33, 46, 59,  72,  85;
  23, 38, 53, 68,  83,  98, 113;
  26, 43, 60, 77,  94, 111, 128, 145;
  29, 48, 67, 86, 105, 124, 143, 162, 181;
  32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

[2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011
From G. C. Greubel, Oct 14 2023: (Start)
T(n, n) = A001844(n).
T(n, n-1) = A001105(n), n >= 2.
T(n, n-2) = A142463(n-1), n >= 3.
T(n, n-3) = (-1)*A147973(n+2), n >= 4.
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

Original entry on oeis.org

0, 3, 8, 6, 13, 20, 9, 18, 27, 36, 12, 23, 34, 45, 56, 15, 28, 41, 54, 67, 80, 18, 33, 48, 63, 78, 93, 108, 21, 38, 55, 72, 89, 106, 123, 140, 24, 43, 62, 81, 100, 119, 138, 157, 176, 27, 48, 69, 90, 111, 132, 153, 174, 195, 216, 30, 53, 76, 99, 122, 145, 168, 191, 214, 237, 260

Offset: 1

Vincenzo Librandi, Jan 25 2009

			Triangle begins:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27, 36;
  12, 23, 34, 45,  56;
  15, 28, 41, 54,  67,  80;
  18, 33, 48, 63,  78,  93, 108;
  21, 38, 55, 72,  89, 106, 123, 140;
  24, 43, 62, 81, 100, 119, 138, 157, 176;
  27, 48, 69, 90, 111, 132, 153, 174, 195, 216;

Vincenzo Librandi, Rows n = 1..100, flattened

All terms are in A155723.
Cf. A008585, A016885, A017053, A020705, A154685, A155722.
Cf. A162261 (row sums).
Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).
Diagonals include: A139570, A181510, A271625.

/* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // Bruno Berselli, Aug 31 2012

Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* Vincenzo Librandi, Mar 01 2012 *)

def A155724(n,k): return 2*n*k+n+k-4
print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

T(n, k) = A154685(n, k) - 8. - L. Edson Jeffery, Oct 12 2012
2*T(n, k) + 9 = (2*k+1)*(2*n+1). - Vincenzo Librandi, Nov 18 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*( 4*(-1)^(n+1)*n^2 + 2*(2-3*(-1)^n)*n - 7*(1-(-1)^n)).
G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)

Edited by N. J. A. Sloane, Jun 23 2010

A013824 a(n) = 2^(5*n + 3).

Original entry on oeis.org

8, 256, 8192, 262144, 8388608, 268435456, 8589934592, 274877906944, 8796093022208, 281474976710656, 9007199254740992, 288230376151711744, 9223372036854775808, 295147905179352825856, 9444732965739290427392, 302231454903657293676544, 9671406556917033397649408

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Cf. A000079 (2^n), A009976, A013822, A013823, A016885 (5*n+3).

[2^(5*n+3): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

2^(5*Range[0, 20] + 3) (* Paolo Xausa, Feb 21 2025 *)

a(n)=2^(5*n+3) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 32*a(n-1), n > 0; a(0)=8.
G.f.: 8/(1-32*x).
a(n) = 8*A009976(n). (End)
From Elmo R. Oliveira, Feb 20 2025: (Start)
E.g.f.: 8*exp(32*x).
a(n) = A000079(A016885(n)). (End)
a(n) = 2*A013823(n) = 4*A013822(n). - Paolo Xausa, Feb 21 2025

A335365 Numbers that are unreachable by the process of starting from 1 and adding 5 and/or multiplying by 3.

Original entry on oeis.org

2, 4, 5, 7, 10, 12, 15, 17, 20, 22, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285

Offset: 1

Alonso del Arte, Jun 03 2020

Start with 1. Add 5 or multiply by 3. Then either add 5 or multiply by 3, and so on and so forth. Following both branches at each step, we can create a tree like this:
                                       1
                    ................../ \..................
                   6                                       3
        11......../ \........18                  8......../ \........9
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       33         23       54         13       24         14       27
  21  48   38  99     28  69   59  162    18  39   29  72     19  42   32  81
According to Haverbeke (2019), some numbers, like 13, are reachable by this process in at least one way. Other numbers, like 15, are completely unreachable.
In fact, almost all positive integers that are not multiples of 5 are reachable, and all multiples of 5 (A008587) are unreachable.
The latter assertion is proven easily enough by taking note of the powers of 3 modulo 5: 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 2, ... (A070352).
As for the former assertion, it is enough to note that 26, 27, 28 and 29 are reachable. Given 5k + r, with k > 4 and r one of 1, 2, 3, 4, start with the solution for 25 + r and then, k - 5 times, add 5.
More precisely the sequence consists of all multiples of 5, numbers less than 25 congruent to 2 (mod 5), and 4. - M. F. Hasler, Jun 05 2020

			Starting with 1, either adding 5 or multiplying by 3 results in a number greater than 2, so 2 is unreachable and therefore in the sequence.
Starting with 1, multiplying by 3 gives 3, proving 3 is reachable and therefore not in the sequence.

Marijn Haverbeke, Eloquent JavaScript, 3rd Ed. San Francisco (2019): No Starch, p. 51.

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

Cf. A008587 (subset), A070352, A335392.

Subsets of the complement: A000244, A016861, A016873 (except for first five terms), A016885, A016897 (except for 4).

JavaScript
```
// See Haverbeke (2019).
```

LinearRecurrence[{2,-1},{2,4,5,7,10,12,15,17,20,22,25,30},70] (* Harvey P. Dale, Apr 01 2023 *)

{is(n)=!(n%5&& !while(n>4, n%3|| is(n/3)|| break (n=1); n-=5)&& n%2==1)} \\ Using exhaustive search, for illustration. - M. F. Hasler, Jun 05 2020

select( {is(n)=n%5==0|| (n<23&&(n%5==2||n==4))}, [1..199]) \\ Much more efficient. - M. F. Hasler, Jun 05 2020

Vec(x*(2 - x^2 + x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 + 2*x^11) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Jun 07 2020

// Based on Haverbeke (2019)
def find153Sol(n: Int): List[Int] = {
  def recur153(curr: Int, history: List[Int]): List[Int] = {
    if (curr == n) history.drop(1) :+ n else if (curr > n) List() else {
      val add5Branch = recur153(curr + 5, history :+ curr)
      if (add5Branch.nonEmpty) add5Branch
          else recur153(curr * 3, history :+ curr)
    }
  }
  recur153(1, List(1))
}
(1 to 200).filter(find153Sol(_).isEmpty)

G.f.: (2*x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 + x^3 - x^2 + 2)*x/(x - 1)^2. - Alois P. Heinz, Jun 05 2020
From Colin Barker, Jun 07 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n>12.
a(n) = 5*(n-6) for n>10.
(End)

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

Original entry on oeis.org

-5, -2, 3, 1, 8, 15, 4, 13, 22, 31, 7, 18, 29, 40, 51, 10, 23, 36, 49, 62, 75, 13, 28, 43, 58, 73, 88, 103, 16, 33, 50, 67, 84, 101, 118, 135, 19, 38, 57, 76, 95, 114, 133, 152, 171, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 25, 48, 71, 94, 117, 140, 163, 186, 209

Offset: 1

Vincenzo Librandi, Jan 24 2009

The numbers 2*T(m,n)+19 =(2*n+1)*(2*m+1) are not prime.

First column: A016777, second column: A016885, third column: A016993, fourth column: A017209. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
-5;
-2, 3;
1,  8,  15;
4,  13, 22, 31;
7,  18, 29, 40, 51;
10, 23, 36, 49, 62,  75;
13, 28, 43, 58, 73,  88,  103;
16, 33, 50, 67, 84,  101, 118, 135;
19, 38, 57, 76, 95,  114, 133, 152, 171;
22, 43, 64, 85, 106, 127, 148, 169, 190, 211; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153143, A153144, A016777, A016885, A016993, A017209.

[2*n*k + n + k - 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k - 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A016886 a(n) = (5*n + 3)^2.

Original entry on oeis.org

9, 64, 169, 324, 529, 784, 1089, 1444, 1849, 2304, 2809, 3364, 3969, 4624, 5329, 6084, 6889, 7744, 8649, 9604, 10609, 11664, 12769, 13924, 15129, 16384, 17689, 19044, 20449, 21904, 23409, 24964, 26569, 28224, 29929, 31684, 33489, 35344, 37249, 39204, 41209

Offset: 0

N. J. A. Sloane

			a(0) = (5*0 + 3)^2 = 9.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Polygamma Function.
Wikipedia, Polygamma Function.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A016885.

[(5*n + 3)^2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 02 2021

(5*Range[0,40]+3)^2 (* or *) LinearRecurrence[{3,-3,1},{9,64,169},40] (* Harvey P. Dale, Dec 09 2016 *)
CoefficientList[Series[(9 + x) (1 + 4 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, Mar 29 2017 *)

Vec((9 + x)*(1 + 4*x) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Mar 29 2017

From Colin Barker, Mar 29 2017: (Start)
G.f.: (9 + x)*(1 + 4*x) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
(End)
a(n) = A000290(A016885(n)). - Michel Marcus, Mar 30 2017
Sum_{n>=0} 1/a(n) = polygamma(1, 3/5)/25. - Amiram Eldar, Oct 02 2020

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

Original entry on oeis.org

10, 13, 18, 16, 23, 30, 19, 28, 37, 46, 22, 33, 44, 55, 66, 25, 38, 51, 64, 77, 90, 28, 43, 58, 73, 88, 103, 118, 31, 48, 65, 82, 99, 116, 133, 150, 34, 53, 72, 91, 110, 129, 148, 167, 186, 37, 58, 79, 100, 121, 142, 163, 184, 205, 226, 40, 63, 86, 109, 132, 155, 178

Offset: 1

Vincenzo Librandi, Aug 02 2009

The numbers 2*T(n,m)-11 = (2*n+1)*(2*m+1) are not prime, and 2*T(n,n) = (2n+1)^2.

First column: A112414, second column: A016885, third column: A017005, fourth column: A017173. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  10;
  13, 18;
  16, 23, 30;
  19, 28, 37, 46;
  22, 33, 44, 55,  66;
  25, 38, 51, 64,  77,  90;
  28, 43, 58, 73,  88,  103, 118;
  31, 48, 65, 82,  99,  116, 133, 150;
  34, 53, 72, 91,  110, 129, 148, 167, 186;
  37, 58, 79, 100, 121, 142, 163, 184, 205, 226;
  40, 63, 86, 109, 132, 155, 178, 201, 224, 247, 270;
  etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153045, A154685, A163657, A112414, A016885, A017005, A017173.

[2*n*k + n + k + 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

T(n,m) = A154685(n,m)+2 = A163657(n,m)-2. [R. J. Mathar, Oct 22 2009]

Comment clarified by R. J. Mathar, Oct 22 2009

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).

Original entry on oeis.org

0, 12, 64, 156, 288, 460, 672, 924, 1216, 1548, 1920, 2332, 2784, 3276, 3808, 4380, 4992, 5644, 6336, 7068, 7840, 8652, 9504, 10396, 11328, 12300, 13312, 14364, 15456, 16588, 17760, 18972, 20224, 21516, 22848, 24220, 25632, 27084, 28576, 30108, 31680, 33292, 34944

Offset: 0

Omar E. Pol, Aug 19 2011

Even 12-gonal numbers. Bisection of A051624.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016885, A051624, A147874.

PolygonalNumber[12,2*Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 02 2017 *)

a(n)=4*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A147874(n+1).
a(n) = 4*n*A016885(n-1), n >= 1.
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 4*x*(3 + 7*x)/(1 - x)^3.
E.g.f.: 4*x*(3 + 5*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A226785 If n=0 (mod 7) then a(n)=0, otherwise a(n)=7^(-1) in Z/nZ*.

Original entry on oeis.org

0, 1, 1, 3, 3, 1, 0, 7, 4, 3, 8, 7, 2, 0, 13, 7, 5, 13, 11, 3, 0, 19, 10, 7, 18, 15, 4, 0, 25, 13, 9, 23, 19, 5, 0, 31, 16, 11, 28, 23, 6, 0, 37, 19, 13, 33, 27, 7, 0, 43, 22, 15, 38, 31, 8, 0, 49, 25, 17, 43, 35, 9, 0

Offset: 1

José María Grau Ribas, Jun 18 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1).

Cf. A092092, A226782-A226787.

A226785 := proc(n)
    local x,a,m ;
    a := 7 ;
    m := 7 ;
    if igcd(n,m) > 1 or n =1 then
        0;
    else
        msolve(x*a=1,n) ;
        op(%) ;
        op(2,%) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[7, n],{n, 1, 122}]
Join[{0},LinearRecurrence[{0,0,0,0,0,0,2,0,0,0,0,0,0,-1},{1,1,3,3,1,0,7,4,3,8,7,2,0,13},70]] (* Harvey P. Dale, Nov 15 2014 *)
Table[If[Mod[n, 7]==0, 0, ModularInverse[7, n]], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)

a(n)=if(n%7,lift(Mod(1,n)/7),0) \\ Charles R Greathouse IV, Jun 18 2013

G.f.: -x^2*(x^13 -x^10 -2*x^9 -x^8 -2*x^7 -7*x^6 -x^4 -3*x^3 -3*x^2 -x -1) / (x^14 -2*x^7 +1). a(n) = 2*a(n-7)-a(n-14). - Colin Barker, Jun 20 2013

a(7n+1) = 6*n+1, n>0. a(7n+2)=A016777(n). a(7n+3) = A005408(n). a(7n+4) = A016885(n). a(7n+5)= A004767(n). a(7n+6)= n+1. - R. J. Mathar, Jun 28 2013

cp's OEIS Frontend

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A155724 Triangle read by rows: T(n, k) = 2*n*k + n + k - 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

Extensions

A013824 a(n) = 2^(5*n + 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A335365 Numbers that are unreachable by the process of starting from 1 and adding 5 and/or multiplying by 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

JavaScript

Mathematica

PARI

PARI

PARI

Scala

Formula

A155551 Triangle read by rows where T(m,n)=2*m*n + m + n - 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A016886 a(n) = (5*n + 3)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A163652 Triangle read by rows where T(n,m)=2*m*n + m + n + 6.

Views

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.