A017257 -id:A017257 - cp's OEIS frontend

A017173 a(n) = 9*n + 1.

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478

Offset: 0

N. J. A. Sloane

Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009
A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009
If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093644 ((9,1) Pascal, column m=1).
Cf. A010701, A010888, A016777, A116371, A147296, A156144, A156145.
Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).

a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..]  -- Reinhard Zumkeller, Feb 04 2014

Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{1,10},60] (* Harvey P. Dale, Dec 27 2014 *)

forstep(n=1,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

[i+1 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

G.f.: (1 + 8*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010
E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023
a(n) = A016777(3*n). - Elmo R. Oliveira, Apr 12 2025

A002282 a(n) = 8*(10^n - 1)/9.

Original entry on oeis.org

0, 8, 88, 888, 8888, 88888, 888888, 8888888, 88888888, 888888888, 8888888888, 88888888888, 888888888888, 8888888888888, 88888888888888, 888888888888888, 8888888888888888, 88888888888888888, 888888888888888888, 8888888888888888888, 88888888888888888888, 888888888888888888888

Offset: 0

N. J. A. Sloane

If the initial term is omitted, might be called eightful (or hateful) numbers!

			Curious multiplications:
9*9 + 7 = 88;
98*9 + 6 = 888;
987*9 + 5 = 8888;
9876*9 + 4 = 88888;
98765*9 + 3 = 888888;
987654*9 + 2 = 8888888;
9876543*9 + 1 = 88888888;
98765432*9 + 0 = 888888888;
987654321*9 - 1 = 8888888888;
9876543210*9 - 2 = 88888888888. - _Philippe Deléham_, Mar 09 2014

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 32.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002283, A059988, A075412, A178635.

Cf. A010785, A017257, A051003, A059482, A246058.

A002282:=n->8*(10^n - 1)/9; seq(A002282(n), n=0..20); # Wesley Ivan Hurt, Mar 10 2014

LinearRecurrence[{11,-10}, {0,8}, 20] (* Harvey P. Dale, May 30 2013 *)

{ a=-4/5; for (n = 0, 200, a+=8*10^(n - 1); write("b002282.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

def a(n): return 8*(10**n - 1)//9 # Martin Gergov, Oct 19 2022

From Jaume Oliver Lafont, Feb 03 2009: (Start)
a(n) = 11*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=8.
G.f.: 8*x/((1-x)*(1-10*x)). (End)
a(n) = A178635(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 8*10^(n-1), with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = 8*A002275(n) = A002283(n) - A002275(n). - Carauleanu Marc, Sep 03 2016
From Ilya Gutkovskiy, Sep 03 2016: (Start)
E.g.f.: 8*(exp(9*x) - 1)*exp(x)/9.
a(n) = floor(8*10^n/9). (End)
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A246058(n) - 1)/2.
a(n) = A010785(A017257(n-1)) for n >= 1. (End)

A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n(9n+7)/2.

Original entry on oeis.org

0, 8, 25, 51, 86, 130, 183, 245, 316, 396, 485, 583, 690, 806, 931, 1065, 1208, 1360, 1521, 1691, 1870, 2058, 2255, 2461, 2676, 2900, 3133, 3375, 3626, 3886, 4155, 4433, 4720, 5016, 5321, 5635, 5958, 6290, 6631, 6981, 7340, 7708, 8085, 8471, 8866, 9270

Offset: 0

Floor van Lamoen, Jul 21 2001

Old name: Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,8,...

Sequence found by reading the line from 0, in the direction 0, 25, ... and the line from 8, in the direction 8, 51, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Jul 24 2012

			The spiral begins:
          15
          / \
        16  14
        /     \
      17   3  13
      /   / \   \
    18   4   2  12
    /   /     \   \
  19   5   0---1  11
  /   /             \
20   6---7---8---9--10

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

Cf. A051682.

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, this sequence, A135705.

List([0..50], n-> n*(9*n+7)/2); # G. C. Greubel, May 24 2019

[n*(9*n+7)/2: n in [0..50]]; // G. C. Greubel, May 24 2019

Table[n*(9*n+7)/2, {n,0,50}] (* G. C. Greubel, May 24 2019 *)
LinearRecurrence[{3,-3,1},{0,8,25},50] (* Harvey P. Dale, Sep 06 2019 *)

a(n)=n*(9*n+7)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(9*n+7)/2 for n in (0..50)] # G. C. Greubel, May 24 2019

a(n) = n*(9*n+7)/2.
a(n) = 9*n + a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
From Bruno Berselli, Jan 13 2011: (Start)
G.f.: x*(8 + x)/(1 - x)^3.
a(n) = Sum_{i=0..n-1} A017257(i) for n > 0. (End)
a(n) = A218470(9n+7). - Philippe Deléham, Mar 27 2013
E.g.f.: x*(16 + 9*x)*exp(x)/2. - G. C. Greubel, May 24 2019

New name from Bruno Berselli (with the original formula), Jan 13 2011

A188866 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

Original entry on oeis.org

2, 4, 3, 8, 7, 4, 16, 17, 10, 5, 32, 41, 26, 13, 6, 64, 99, 68, 35, 16, 7, 128, 239, 178, 95, 44, 19, 8, 256, 577, 466, 259, 122, 53, 22, 9, 512, 1393, 1220, 707, 340, 149, 62, 25, 10, 1024, 3363, 3194, 1931, 950, 421, 176, 71, 28, 11, 2048, 8119, 8362, 5275, 2658, 1193, 502, 203, 80, 31, 12

Offset: 1

R. H. Hardin, Apr 12 2011

Number of 0..n strings of length k and adjacent elements differing by one or less. (See link for bijection.) Equivalently, number of base (n+1) k digit numbers with adjacent digits differing by one or less. - Andrew Howroyd, Mar 30 2017
All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions. - Andrew Howroyd, Apr 15 2017
Equivalently, the number of walks of length k-1 on the path graph P_{n+1} with a loop added at each vertex. - Pontus von Brömssen, Sep 08 2021

			Table starts:
   2  4  8  16  32   64  128   256   512   1024   2048    4096    8192    16384
   3  7 17  41  99  239  577  1393  3363   8119  19601   47321  114243   275807
   4 10 26  68 178  466 1220  3194  8362  21892  57314  150050  392836  1028458
   5 13 35  95 259  707 1931  5275 14411  39371 107563  293867  802859  2193451
   6 16 44 122 340  950 2658  7442 20844  58392 163594  458356 1284250  3598338
   7 19 53 149 421 1193 3387  9627 27383  77923 221805  631469 1797957  5119593
   8 22 62 176 502 1436 4116 11814 33942  97582 280676  807574 2324116  6689624
   9 25 71 203 583 1679 4845 14001 40503 117263 339699  984515 2854281  8277153
  10 28 80 230 664 1922 5574 16188 47064 136946 398746 1161634 3385486  9869934
  11 31 89 257 745 2165 6303 18375 53625 156629 457795 1338779 3916897 11463989
Some solutions for 5 X 3:
  1 1 1   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1   1 1 1
  1 1 1   0 0 1   0 1 1   1 1 1   0 0 0   1 0 0   1 0 1
  0 0 0   0 0 0   0 0 1   1 1 1   0 0 0   0 0 0   0 0 0
  0 0 0   0 0 0   0 0 0   1 1 0   0 0 0   0 0 0   0 0 0
  0 0 0   0 0 0   0 0 0   0 0 0   0 0 0   0 0 0   0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..1741
Andrew Howroyd, Bijection with 0..n strings of length k.
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Columns 2..8 are A016777, A017257(n-1), A188861-A188865.
Rows 2..31 are A001333(n+1), A126358, A057960(n+1), A126360, A002714, A126362-A126386.
Main diagonal is A188860.
Cf. A276562, A220062.

rows = 11; rowGf[n_, x_] = 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 + ChebyshevU[n-1, (1-x)/(2*x)])/ChebyshevU[n, (1-x)/(2*x)])/(1-3*x)^2;
row[n_] := rowGf[n+1, x] + O[x]^(rows+1) // CoefficientList[#, x]& // Rest; T = Array[row, rows]; Table[T[[n-k+1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)

\\ from Knopfmacher et al.
RowGf(k, x='x) = my(z=(1-x)/(2*x)); 1 + (x*(k-(3*k+2)*x) + (2*x^2)*(1+polchebyshev(k-1, 2, z))/polchebyshev(k, 2, z))/(1-3*x)^2;
T(n,k) = {polcoef(RowGf(n+1) + O(x*x^k),k)}
for(n=1, 10, print(Vec(RowGf(n+1) + O(x^11)))) \\ Andrew Howroyd, Apr 15 2017 [updated Mar 13 2021]

Empirical: T(n,1) = n + 1.
Empirical: T(n,2) = 3*n + 1.
Empirical: T(n,3) = 9*n - 1.
Empirical: T(n,4) = 27*n - 13 for n > 1.
Empirical: T(n,5) = 81*n - 65 for n > 2.
Empirical: T(n,6) = 243*n - 265 for n > 3.
Empirical: T(n,7) = 729*n - 987 for n > 4.
Empirical: T(n,8) = 2187*n - 3495 for n > 5.
Empirical: T(1,k) = 2*T(1,k-1).
Empirical: T(2,k) = 2*T(2,k-1) + T(2,k-2).
Empirical: T(3,k) = 3*T(3,k-1) - T(3,k-2).
Empirical: T(4,k) = 3*T(4,k-1) - 2*T(4,k-3).
Empirical: T(5,k) = 4*T(5,k-1) - 3*T(5,k-2) -   T(5,k-3).
Empirical: T(6,k) = 4*T(6,k-1) - 2*T(6,k-2) - 4*T(6,k-3) +   T(6,k-4).
Empirical: T(7,k) = 5*T(7,k-1) - 6*T(7,k-2) -   T(7,k-3) + 2*T(7,k-4).
Empirical: T(8,k) = 5*T(8,k-1) - 5*T(8,k-2) - 5*T(8,k-3) + 5*T(8,k-4) + T(8,k-5).

A221596 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less.

Original entry on oeis.org

0, 0, 4, 0, 7, 8, 0, 10, 17, 16, 0, 13, 26, 49, 32, 0, 16, 35, 100, 139, 64, 0, 19, 44, 169, 342, 393, 128, 0, 22, 53, 256, 651, 1210, 1113, 256, 0, 25, 62, 361, 1068, 2715, 4240, 3151, 512, 0, 28, 71, 484, 1593, 5082, 11011, 14898, 8921, 1024, 0, 31, 80, 625, 2226, 8475

Offset: 1

R. H. Hardin Jan 20 2013

Table starts
....0......0.......0........0........0.........0.........0.........0..........0
....4......7......10.......13.......16........19........22........25.........28
....8.....17......26.......35.......44........53........62........71.........80
...16.....49.....100......169......256.......361.......484.......625........784
...32....139.....342......651.....1068......1593......2226......2967.......3816
...64....393....1210.....2715.....5082......8475.....13056.....18987......26430
..128...1113....4240....11011....22912.....41401.....67936....103975.....150976
..256...3151...14898....45099...105586....210101....374342....615965.....954572
..512...8921...52306...184063...482204...1047967...2006006...3504371....5714456
.1024..25257..183684...752155..2210256...5267759..10894988..20352239...35218688
.2048..71507..645006..3072247.10115926..26387005..58789204.116958723..213700742
.4096.202449.2264978.12550859.46327024.132384353.318224626.675761541.1307528098

			Some solutions for n=6 k=4
..0....2....3....3....2....2....1....1....2....4....4....2....0....2....4....1
..0....1....4....2....3....3....2....1....3....3....4....2....1....3....3....2
..2....4....4....2....1....0....2....3....4....0....4....1....0....2....4....2
..2....4....1....4....1....0....1....2....0....1....3....4....0....2....4....1
..0....1....1....3....0....2....2....3....1....4....3....3....1....1....0....0
..0....2....1....2....1....3....1....2....2....3....2....3....0....1....0....0

R. H. Hardin, Table of n, a(n) for n = 1..2080

Column 3 is A221568
Row 2 is A016777
Row 3 is A017257(n-1)
Row 4 is A016778

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 2*a(n-1) +2*a(n-2) +a(n-3) for n>4
k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
k=4: a(n) = 3*a(n-1) +4*a(n-2) +6*a(n-4) +4*a(n-5) +4*a(n-6)
k=5: a(n) = 4*a(n-1) +3*a(n-2) -6*a(n-3) +19*a(n-4) +5*a(n-5) +a(n-6)
k=6: a(n) = 4*a(n-1) +5*a(n-2) -7*a(n-3) +33*a(n-4) +17*a(n-5) +24*a(n-6) -5*a(n-7) +2*a(n-8)
k=7: a(n) = 5*a(n-1) +3*a(n-2) -16*a(n-3) +65*a(n-4) -14*a(n-5) +23*a(n-6) +2*a(n-7) +8*a(n-8)
Empirical for row n:
n=2: a(n) = 3*n + 1
n=3: a(n) = 9*n - 1
n=4: a(n) = 9*n^2 + 6*n + 1
n=5: a(n) = 54*n^2 - 69*n + 63 for n>2
n=6: a(n) = 27*n^3 + 108*n^2 - 252*n + 267 for n>3
n=7: a(n) = 243*n^3 - 351*n^2 + 237*n + 127 for n>2

A235332 a(n) = n(9n + 25)/2 + 6.

Original entry on oeis.org

6, 23, 49, 84, 128, 181, 243, 314, 394, 483, 581, 688, 804, 929, 1063, 1206, 1358, 1519, 1689, 1868, 2056, 2253, 2459, 2674, 2898, 3131, 3373, 3624, 3884, 4153, 4431, 4718, 5014, 5319, 5633, 5956, 6288, 6629, 6979, 7338, 7706, 8083, 8469, 8864, 9268, 9681, 10103

Offset: 0

Bruno Berselli, Jan 22 2014

This is the case d=6 of n*(9*n + 4*d + 1)/2 + d. Other similar sequences are:
d=0, A022267;
d=1, A064225;
d=2, A062123;
d=3, A064226;
d=4, A022266 (with initial 0);
d=5, A178977.
First bisection of A235537.

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

Cf. A017257 (first differences), A022266, A022267, A062123, A064225, A064226, A081267, A178977, A235537.

Magma
```
[n*(9*n+25)/2+6: n in [0..50]];
```

Table[n (9 n + 25)/2 + 6, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{6,23,49},50] (* Harvey P. Dale, Feb 12 2022 *)

a(n)=n*(9*n+25)/2+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (6 + 5*x - 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
2*a(n) - a(n+1) + 12 = A081267(n).
E.g.f.: exp(x)*(12 + 34*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

A247679 Composite numbers congruent to 17 modulo 18.

Original entry on oeis.org

35, 125, 143, 161, 215, 287, 305, 323, 341, 377, 395, 413, 485, 539, 575, 611, 629, 665, 737, 755, 791, 845, 899, 917, 935, 989, 1007, 1025, 1043, 1079, 1115, 1133, 1169, 1205, 1241, 1295, 1313, 1331, 1349, 1385, 1403, 1421, 1457, 1475

Offset: 1

Odimar Fabeny, Sep 22 2014

Subsequence of A017257 (9n + 8).

Odimar Fabeny, Table of n, a(n) for n = 1..10000

Cf. A017257, A247676, A247678, A247681, A247682, A247683.

Select[18Range[100] - 1, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 25 2014 *)
Select[Range[17,1500,18],CompositeQ] (* Harvey P. Dale, Jun 19 2022 *)

lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 8), print1(n, ", ")); ); } \\ Michel Marcus, Sep 22 2014

a(n) ~ 18n. - Charles R Greathouse IV, Sep 27 2014

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

Original entry on oeis.org

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

A155546 Triangle read by rows where T(m,n)=2mn + m + n - 5, 1<=n<=m.

Original entry on oeis.org

-1, 2, 7, 5, 12, 19, 8, 17, 26, 35, 11, 22, 33, 44, 55, 14, 27, 40, 53, 66, 79, 17, 32, 47, 62, 77, 92, 107, 20, 37, 54, 71, 88, 105, 122, 139, 23, 42, 61, 80, 99, 118, 137, 156, 175, 26, 47, 68, 89, 110, 131, 152, 173, 194, 215, 29, 52, 75, 98, 121, 144, 167, 190, 213

Offset: 1

Vincenzo Librandi, Jan 24 2009

2*T(m,n)+11 = (2*m+1)*(2*n+1) is not prime.

First column: A016789, second column: A016873, third column: A017041, fourth column: A017257. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
-1;
2,  7;
5,  12, 19;
8,  17, 26, 35;
11, 22, 33, 44, 55;
14, 27, 40, 53, 66,  79;
17, 32, 47, 62, 77,  92,  107;
20, 37, 54, 71, 88,  105, 122, 139;
23, 42, 61, 80, 99,  118, 137, 156, 175;
26, 47, 68, 89, 110, 131, 152, 173, 194, 215; etc.

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

Cf. A101448, A153083, A016789, A016873, A017041, A017257.

[2*n*k + n + k - 5: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

t[n_,k_]:= 2 n*k + n + k - 5; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

A013656 a(n) = n(9n-2).

Original entry on oeis.org

0, 7, 32, 75, 136, 215, 312, 427, 560, 711, 880, 1067, 1272, 1495, 1736, 1995, 2272, 2567, 2880, 3211, 3560, 3927, 4312, 4715, 5136, 5575, 6032, 6507, 7000, 7511, 8040, 8587, 9152, 9735, 10336, 10955, 11592, 12247, 12920, 13611, 14320, 15047, 15792, 16555

Offset: 0

David W. Wilson

For n>0, numbers such that sqrt(a(n)) has the continued fraction {k;[1,1,1,2k]}, where the part in [] is repeated and k is of the form 3m+2 (A016789). - Bruno Berselli, May 30 2013

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n-1; {3, 3n-1, 3, 12n-2}]. - Magus K. Chu, Sep 18 2022

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A010701, A017257, A185019.

Cf. A016789, A061039, A144454.

[n*(9*n-2): n in [0..60]]; // G. C. Greubel, Mar 11 2022

Table[n(9n-2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,32},50] (* Harvey P. Dale, Jul 07 2012 *)

a(n)=n*(9*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n+1) = A144454(9*n+7) = A061039(27*n+21). - Paul Curtz, Nov 05 2008
a(n) = a(n-1) + 18*n - 11 with n>0, a(0)=0. - Vincenzo Librandi, Nov 22 2010
a(0)=0, a(1)=7, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 07 2012
From G. C. Greubel, Mar 11 2022: (Start)
G.f.: x*(7 - 11*x)/(1-x)^3.
E.g.f.: x*(7 + 9*x)*exp(x). (End)
Sum_{n>=1} 1/a(n) = -(psi(7/9)+gamma)/2 = (A354640-A001620)/2 = 0.22000753... - R. J. Mathar, Apr 22 2024

cp's OEIS Frontend

A017173 a(n) = 9*n + 1.

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A002282 a(n) = 8*(10^n - 1)/9.

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A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n+7)/2.

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A188866 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

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A221596 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less.

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A235332 a(n) = n*(9*n + 25)/2 + 6.

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A247679 Composite numbers congruent to 17 modulo 18.

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A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n(9n+7)/2.

A235332 a(n) = n(9n + 25)/2 + 6.

A013656 a(n) = n(9n-2).