A017173 -id:A017173 - cp's OEIS frontend

A090570 Numbers that are congruent to {0, 1} mod 9.

0, 1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99, 100, 108, 109, 117, 118, 126, 127, 135, 136, 144, 145, 153, 154, 162, 163, 171, 172, 180, 181, 189, 190, 198, 199, 207, 208, 216, 217, 225, 226

Offset: 1

Giovanni Teofilatto, Feb 25 2004

			13 is 1101 in base 2, so a(13+1) = a(14) = 36*1 + 18*1 + 9*0 + 1*1 = 36+18+1 = 55. - _Philippe Deléham_, Oct 17 2011

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

Union of A008591 and A017173. - Reinhard Zumkeller, Oct 10 2008
Cf. A010888, A145389.
Cf. A005010, A030308.

forstep(n=0,200,[1,8],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

A145389(a(n)) = A010888(a(n)). - Reinhard Zumkeller, Oct 10 2008
a(n) = 9*n - a(n-1) - 17 (with a(1)=0). - Vincenzo Librandi, Nov 16 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 9*n/2 - 25/4 - 7*(-1)^n/4.
G.f.: x^2*(1+8*x)/( (1+x)*(1-x)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A005010(k-1), with A005010(-1)=1. - Philippe Deléham, Oct 17 2011.
E.g.f.: 8 + ((18*x - 25)*exp(x) - 7*exp(-x))/4. - David Lovler, Sep 03 2022

A213651 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 282, 348, 415, 480

Offset: 0

M. F. Hasler, Jun 17 2012

The n-th row also yields the number of ways to get a total of n, n+1, ..., 10n, when throwing n 10-sided dice, or summing n integers ranging from 1 to 10.
The row sums equal 10^n = A011557(n).
The row lengths are 1 + 9n = 10n - (n-1) = A017173(n).
T(n,k) is the number of integers in the [0, 10^n-1] range distributed according to the sum k of their digits. - Miquel Cerda, Jun 21 2017
The sum of the squares of the integers of the n-th row gives A174061(n). - Miquel Cerda, Jul 03 2017

			There are 1, 3, 6, 10, ... ways to score a total of 4, 5, 6, 7, ... when throwing three 10-sided dice.
The table begins as follows:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1,1; (row sum = 10, row length = 10)
(row n=2) 1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1; (sum = 100, length = 19)
(row n=3) 1,3,6,10,15,21,28,36,45,55,63,69,73,75,75,73,...; row sum = 1000;
(row n=4) 1,4,10,20,35,56,84,120,165,220,282,348,415,...; row sum = 10^4;
etc.
Number of integers in (row n=2): k(2)=3, because in the range 0 to 99 there are 3 integers whose digits sum to 2: 2, 11 and 20. - _Miquel Cerda_, Jun 21 2017

Seiichi Manyama, Rows n = 0..46, flattened
Miquel Cerda, Graphical construction of the triangle T(n,k) for n = 0..11.

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 10-nomials as a table
r := 10:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013

concat(vector(5,k,Vec(sum(j=0,9,x^j)^(k-1))))

T(n,k) = Sum_{i = 0..floor(k/10)} (-1)^i*binomial(n,i)*binomial(n+k-1-10*i,n-1) for n >= 0 and 0 <= k <= 9*n. - Peter Bala, Sep 07 2013

A081045 10th binomial transform of (1,9,0,0,0,0,0,...).

Original entry on oeis.org

1, 19, 280, 3700, 46000, 550000, 6400000, 73000000, 820000000, 9100000000, 100000000000, 1090000000000, 11800000000000, 127000000000000, 1360000000000000, 14500000000000000, 154000000000000000, 1630000000000000000, 17200000000000000000, 181000000000000000000

Offset: 0

Paul Barry, Mar 04 2003

From Bernard Schott, Nov 12 2022: (Start)
For n >= 1, a(n-1) is the number of digits 1 (or any nonzero digit) that are necessary to write all the n-digit integers, while the corresponding number of digits 0 to write all these n-digit integers is A212704(n-1) for n >=2.
E.g.: a(2-1) = 19 since 19 digits 2's are required to write integers with a digit 2 from 10 up to 99: {12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92}.
First difference of A053541. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A053541, A081044, A081043, A212704.

[(9*n+10)*10^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - x)/(1 - 10 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20,-100},{1,19},20] (* Harvey P. Dale, Dec 28 2023 *)

a(n) = 20*a(n-1) - 100*a(n-2); a(0)=1, a(1)=19.
a(0)=1; for  n>= 1, a(n) = (9*n+10)*10^(n-1) = 10^(n-1)*A017173(n+1).
a(n) = Sum_{k=0..n} (k+1)*9^k*binomial(n, k).
G.f.: (1-x)/(1-10*x)^2.
a(n) = A053541(n+1) - A053541(n), for n >= 1. - Bernard Schott, Nov 12 2022
E.g.f.: exp(10*x)*(1 + 9*x). - Stefano Spezia, Jan 31 2025

A247681 Odd nonprimes congruent to 1 modulo 9.

Original entry on oeis.org

1, 55, 91, 145, 217, 235, 253, 289, 325, 343, 361, 415, 451, 469, 505, 559, 595, 649, 667, 685, 703, 721, 775, 793, 847, 865, 901, 955, 973, 1027, 1045, 1081, 1099, 1135, 1189, 1207, 1225, 1243, 1261, 1315, 1333, 1351, 1369, 1387, 1405

Offset: 1

Odimar Fabeny, Sep 22 2014

Subsequence of A017173.

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

Cf. A017173, A247676, A247678, A247679, A247682, A247683, A247900 (first differences).

Select[18Range[0, 99] + 1, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 25 2014 *)
Select[Range[1,1500,18],!PrimeQ[#]&] (* Harvey P. Dale, Mar 07 2018 *)

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

A156144 Number of partitions of n into parts having in decimal representation the same digital root as n has.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 3, 1, 1, 2, 1, 1, 3, 5, 2, 5, 1, 1, 2, 1, 1, 5, 8, 4, 8, 2, 1, 4, 1, 1, 7, 13, 5, 13, 2, 2, 5, 1, 1, 11, 20, 9, 19, 3, 2, 9, 1, 1, 15, 31, 12, 29, 4, 3, 11, 2, 1, 22, 46, 20, 42, 7, 4, 18, 2, 2, 30, 68, 27, 61, 9, 6, 23, 3, 2, 42, 98, 42, 85

Offset: 1

Reinhard Zumkeller, Feb 05 2009

a(n) <= a(n+9); Max{n: a(n)=1} = 71;
A156145 and A017173 give record values and where they occur: a(A017173(n-1))=A156145(n);
a(A017173(n)) = A116371(A017173(n)).

			a(19) = #{19, 10+1+1+1+1+1+1+1+1+1, 19x1} = 3;
a(20) = #{20, 2+2+2+2+2+2+2+2+2+2} = 2;
a(21) = #{21, 3+3+3+3+3+3+3, 12+3+3+3} = 3;
a(22) = #{22} = 1;

R. Zumkeller, Table of n, a(n) for n = 1..500

A010888, A114102.

a156144 n = p [x | x <- [1..n], a010888 x == a010888 n] n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 04 2014

A172171 (1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

Original entry on oeis.org

1, 1, 10, 1, 11, 19, 1, 12, 30, 28, 1, 13, 42, 58, 37, 1, 14, 55, 100, 95, 46, 1, 15, 69, 155, 195, 141, 55, 1, 16, 84, 224, 350, 336, 196, 64, 1, 17, 100, 308, 574, 686, 532, 260, 73, 1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82

Offset: 1

Mark Dols, Jan 28 2010

Binomial transform of A017173.

			Triangle begins:
  1;
  1, 10;
  1, 11,  19;
  1, 12,  30,  28;
  1, 13,  42,  58,   37;
  1, 14,  55, 100,   95,   46;
  1, 15,  69, 155,  195,  141,   55;
  1, 16,  84, 224,  350,  336,  196,   64;
  1, 17, 100, 308,  574,  686,  532,  260,   73;
  1, 18, 117, 408,  882, 1260, 1218,  792,  333,   82;
  1, 19, 135, 525, 1290, 2142, 2478, 2010, 1125,  415,  91;
  1, 20, 154, 660, 1815, 3432, 4620, 4488, 3135, 1540, 506, 100;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A007318, A017173, A050489 (central terms), A093644, A139634 (row sums).

T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[n==2 && k==2, 10, T[n-1, k] + 2*T[n-1, k-1] - T[n-2, k-1] - T[n-2, k-2]]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 24 2022 *)

@CachedFunction
def T(n,k):
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (n==2 and k==2): return 10
    else: return T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 24 2022

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(n,1) = 1, T(2,2) = 10, T(n,k) = 0 if k < 1 or if k > n.
Sum_{k=0..n} T(n, k) = A139634(n).
T(2*n-1, n) = A050489(n).

More terms from Philippe Deléham, Dec 25 2013

A147296 a(n) = n(9n+2).

Original entry on oeis.org

0, 11, 40, 87, 152, 235, 336, 455, 592, 747, 920, 1111, 1320, 1547, 1792, 2055, 2336, 2635, 2952, 3287, 3640, 4011, 4400, 4807, 5232, 5675, 6136, 6615, 7112, 7627, 8160, 8711, 9280, 9867, 10472, 11095, 11736, 12395, 13072, 13767, 14480, 15211, 15960

Offset: 0

Paul Curtz, Nov 05 2008

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n; {1, 1, 1, 3n-1, 1, 1, 1, 12n}]. - Magus K. Chu, Sep 17 2022

Reply to V. Librandi, A147296 (SeqFan list). - _M. F. Hasler_, Mar 01 2009
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Equals first 9-fold decimation of A144454.

Cf. A010701, A017173, A147296, A185019.

Table[n(9n+2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,40},50] (* Harvey P. Dale, Dec 19 2014 *)

A147296(n) = n*(9*n + 2) \\ M. F. Hasler, Mar 01 2009

a(n) = n*(9*n + 2), as conjectured by V. Librandi. - M. F. Hasler, Mar 01 2009
G.f.: x*(11+7*x)/(1-x)^3. - Jaume Oliver Lafont, Aug 30 2009
a(n) = floor((3*n + 1/3)^2). - Reinhard Zumkeller, Apr 14 2010
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(11 + 9*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

More terms from M. F. Hasler, Mar 01 2009

A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

Original entry on oeis.org

1, 4, 9, 7, 14, 21, 10, 19, 28, 37, 13, 24, 35, 46, 57, 16, 29, 42, 55, 68, 81, 19, 34, 49, 64, 79, 94, 109, 22, 39, 56, 73, 90, 107, 124, 141, 25, 44, 63, 82, 101, 120, 139, 158, 177, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 31, 54, 77, 100, 123, 146, 169

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n)+7 = (2n+1)*(2m+1) is not prime.

First column: A016777; second column: A016897; third column: A008589; fourth column: A017173. - Vincenzo Librandi, Nov 19 2012

			Triangle begins:
1;
4,  9;
7,  14, 21;
10, 19, 28, 37;
13, 24, 35, 46, 57;
16, 29, 42, 55, 68,  81;
19, 34, 49, 64, 79,  94,  109;
22, 39, 56, 73, 90,  107, 124, 141;
25, 44, 63, 82, 101, 120, 139, 158, 177;
28, 49, 70, 91, 112, 133, 154, 175, 196, 217; etc.

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

Cf. A153053, A105760, A162265 (row sums), A016777, A016897, A008589, A017173.

[(2*n*k + n + k - 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 19 2012

t[n_,k_]:=2 n*k + n + k - 3; Table[t[n, k], {n, 20}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 19 2012 *)

A211821 Numbers with all divisors with additive digital root of 1.

Original entry on oeis.org

1, 19, 37, 73, 109, 127, 163, 181, 199, 271, 307, 361, 379, 397, 433, 487, 523, 541, 577, 613, 631, 703, 739, 757, 811, 829, 883, 919, 937, 991, 1009, 1063, 1117, 1153, 1171, 1279, 1297, 1369, 1387, 1423, 1459, 1531, 1549, 1567, 1621, 1657, 1693, 1747, 1783

Offset: 1

Jaroslav Krizek, Apr 26 2012

All divisors of numbers from this sequence are in this sequence. Likewise, the product of any terms in this sequence is a number that is also in this sequence.
Union of A061237 (prime numbers == 1 (mod 9)) and nonprime numbers A211822.
Subsequence of A017173 (numbers of form 9n+1). - Jaroslav Krizek
For prime numbers, it is enough to verify that the number itself is congruent to 1 mod 9. The first composite term is 361, which is the square of the first prime in this sequence. - Alonso del Arte, May 02 2012

			Number 703 with divisors 1, 19, 37, 703 is in sequence because all divisors have additive digital root of 1.

Cf. A211822, A211823, A024906, A061237, A017173.

digitalRoot[n_, b_:10] := FixedPoint[Plus@@IntegerDigits[#, b] &,  n]; A211821 = Select[Range[1, 1999, 9], Union[digitalRoot[Divisors[#]]] == {1} &] (* Alonso del Arte, May 02 2012 *)

a(n) = 9*k(n) + 1 for k(n) = A211823(n).

A301451 Numbers congruent to {1, 7} mod 9.

Original entry on oeis.org

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268

Offset: 1

Bruno Berselli, Mar 21 2018

First bisection of A056991, second bisection of A242660.

The squares of the terms of A174396 are the squares of this sequence.

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

Cf. A274406: numbers congruent to {0, 8} mod 9.
Cf. A193910: numbers congruent to {2, 6} mod 9.
Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;

Magma
```
&cat [[9*n+1, 9*n+7]: n in [0..40]];
```

Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)

Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

[2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]

[n for n in (1..300) if n % 9 in (1,7)]

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.
a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019

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A090570 Numbers that are congruent to {0, 1} mod 9.

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A213651 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...

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A081045 10th binomial transform of (1,9,0,0,0,0,0,...).

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A247681 Odd nonprimes congruent to 1 modulo 9.

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A156144 Number of partitions of n into parts having in decimal representation the same digital root as n has.

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A172171 (1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

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A147296 a(n) = n*(9*n+2).

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A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

A147296 a(n) = n(9n+2).