A017173 -id:A017173 - cp's OEIS frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111

Offset: 0

N. J. A. Sloane

R_n is a string of n 1's.
Base-4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 1. - Paul Barry, Mar 12 2004
Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - Zak Seidov, Dec 05 2008
For n >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012
Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012
The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013
For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014
The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014
From Peter Bala, Sep 20 2015: (Start)
Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)
Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016
If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016
0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016
Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017
The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020
q-integers for q = 10. - John Keith, Apr 12 2021
Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 235-237.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.
Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.
Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.

David Wasserman, Table of n, a(n) for n = 0..1000
Eudes Antonio Costa and Fernando Soares Carvalho, On repunit polynomials sequence, Braz. Elec. J. Math. (2024). See pp. 2, 15.
Eudes Antonio Costa, Douglas Catulio Santos, Paula Maria Machado Cruz Catarino, and Elen Viviani Pereira Spreafico, On Gaussian and Quaternion Repunit Numbers, Rev. Mat. UFOP (Brazil, 2024) Vol. 2. See p. 2.
Eudes Antonio Costa, Paula Maria Machado Cruz Catarino, and Douglas Catulio Santos, A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence, Symmetry (2024) Vol. 17, No. 1, 28.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Douglas Catulio Santos, Eudes Antonio Costa, and Paula Maria Machado Cruz Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
W. M. Snyder, Factoring Repunits, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.
Amelia Carolina Sparavigna, On Repunits, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Repunit.
Eric Weisstein's World of Mathematics, Demlo Number.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Wikipedia, Repunit.
Amin Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6.
Stephen Wolfram, A New Kind of Science.
Samuel Yates, The Mystique of Repunits, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.
Index to Elementary Cellular Automata.
Index entries for 10-automatic sequences.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (11,-10).
Index entries for "core" sequences.

Partial sums of 10^n (A011557). Factors: A003020, A067063.
Bisections give A099814, A100706.
Cf. A000042, A002276, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A004023, A046053, A059988, A065444, A075412, A075415, A083278, A095370, A102380, A125134, A178635, A204845, A204846, A204847, A204848, A206244.
Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.
Cf. A000225, A001019, A001045, A002450, A004215, A015565, A125118, A242614, A242622.
Cf. A010785, A017173, A105279.

a002275 = (`div` 9) . subtract 1 . (10 ^)
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

[(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014

seq((10^k - 1)/9, k=0..30); # Wesley Ivan Hurt, Sep 28 2013

Table[(10^n - 1)/9, {n, 0, 19}] (* Alonso del Arte, Nov 15 2011 *)
Join[{0},Table[FromDigits[PadRight[{},n,1]],{n,20}]] (* Harvey P. Dale, Mar 04 2012 *)

a[0]:0$
a[1]:1$
a[n]:=11*a[n-1]-10*a[n-2]$
A002275(n):=a[n]$
makelist(A002275(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009

my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016

print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022

[lucas_number1(n, 11, 10) for n in range(21)]  # Zerinvary Lajos, Apr 27 2009

a(n) = 10*a(n-1) + 1, a(0)=0.
a(n) = A000042(n) for n >= 1.
Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - Paul Barry, Mar 24 2004
G.f.: x/((1-10*x)*(1-x)). Regarded as base b numbers, g.f. x/((1-b*x)*(1-x)). - Franklin T. Adams-Watters, Jun 15 2006
a(n) = 11*a(n-1) - 10*a(n-2), a(0)=0, a(1)=1. - Lekraj Beedassy, Jun 07 2006
a(n) = A125118(n,9) for n>8. - Reinhard Zumkeller, Nov 21 2006
a(n) = A075412(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = A242614(n,A242622(n)). - Reinhard Zumkeller, Jul 17 2014
E.g.f.: (exp(9*x) - 1)*exp(x)/9. - Ilya Gutkovskiy, May 11 2016
a(n) = Sum_{k=0..n-1} 10^k. - Torlach Rush, Nov 03 2020
Sum_{n>=1} 1/a(n) = A065444. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Aug 02 2025: (Start)
a(n) = A002283(n)/9 = A105279(n)/10.
a(n) = A010785(A017173(n-1)) for n >= 1. (End)

A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5

Offset: 0

N. J. A. Sloane

This is sometimes also called the additive digital root of n.
n mod 9 (A010878) is a very similar sequence.
Partial sums are given by A130487(n-1) + n (for n > 0). - Hieronymus Fischer, Jun 08 2007
Decimal expansion of 13717421/111111111 is 0.123456789123456789123456789... with period 9. - Eric Desbiaux, May 19 2008
Decimal expansion of 13717421 / 1111111110 = 0.0[123456789] (periodic) - Daniel Forgues, Feb 27 2017
a(A005117(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
My friend Jahangeer Kholdi has found that 19 is the smallest prime p such that for each number n, a(p*n) = a(n). In fact we have: a(m*n) = a(a(m)*a(n)) so all numbers with digital root 1 (numbers of the form 9k + 1) have this property. See comment lines of A017173. Also we have a(m+n) = a(a(m) + a(n)). - Farideh Firoozbakht, Jul 23 2010

			The digits of 37 are 3 and 7, and 3 + 7 = 10. And the digits of 10 are 1 and 0, and 1 + 0 = 1, so a(37) = 1.

Martin Gardner, Mathematics, Magic and Mystery, 1956.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digitaddition
Eric Weisstein's World of Mathematics, Digital Root
Wikipedia, Vedic square
Index entries for Colombian or self numbers and related sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A007953, A007954, A031347, A113217, A113218, A010878 (n mod 9), A010872, A010873, A010874, A010875, A010876, A010877, A010879, A004526, A002264, A002265, A002266, A017173, A031286 (additive persistence of n), (multiplicative digital root of n), A031346 (multiplicative persistence of n).

a010888 = until (< 10) a007953
-- Reinhard Zumkeller, Oct 17 2011, May 12 2011

[n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // Bruno Berselli, Mar 18 2016

A010888 := n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # N. J. A. Sloane, Feb 20 2013

Join[{0}, Array[Mod[ # - 1, 9] + 1 &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[Range[0, 1], Table[n - 9 Floor[(n - 1) / 9], {n, 2, 100}]] (* José de Jesús Camacho Medina, Nov 10 2014 *) (* Corrected by Vincenzo Librandi, Nov 11 2014 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 2, 3, 4, 5, 6, 7, 8, 9},104]] (* Ray Chandler, Aug 26 2015 *)
Table[FixedPoint[Total[IntegerDigits[#, 10]] &, n], {n, 0, 104}] (* IWABUCHI Yu(u)ki, Jun 03 2016 *)

A010888(n)=if(n,(n-1)%9+1) \\ M. F. Hasler, Jan 04 2011

def A010888(n):
    return 1 + (n - 1) % 9 if n else 0 # Chai Wah Wu, Aug 23 2014, Apr 23 2023

0 :: List.fill(10)(1 to 9).flatten // Alonso del Arte, Feb 01 2020

If n = 0 then a(n) = 0; otherwise a(n) = (n reduced mod 9), but if the answer is 0 change it to 9.
Equivalently, if n = 0 then a(n) = 0, otherwise a(n) = (n - 1 reduced mod 9) + 1.
If the initial 0 term is ignored, the sequence is periodic with period 9.
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010878(n-1) + 1 (for n > 0).
G.f.: g(x) = x*(Sum_{k = 0..8}(k+1)*x^k)/(1 - x^9). Also: g(x) = x(9x^10 - 10x^9 + 1)/((1 - x^9)(1 - x)^2). (End)
a(n) = n - 9*floor((n-1)/9), for n > 0. - José de Jesús Camacho Medina, Nov 10 2014

A051682 11-gonal (or hendecagonal) numbers: a(n) = n(9n-7)/2.

Original entry on oeis.org

0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, 3683, 3945, 4216, 4496, 4785, 5083, 5390, 5706, 6031, 6365, 6708, 7060, 7421, 7791, 8170

Offset: 0

Barry E. Williams

From Floor van Lamoen, Jul 21 2001: (Start)
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 0,1,...
The spiral begins:
         15
         / \
       16  14
       /     \
     17   3  13
     /   / \   \
   18   4   2  12
       /     \   \
      5   0---1  11
     /             \
    6---7---8---9--10
. (End)
(1), (4+7), (7+10+13), (10+13+16+19), ... - Jon Perry, Sep 10 2004
This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011
Sequence found by reading the line from 0, in the direction 0, 11, ... and the parallel line from 1, in the direction 1, 30, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Jul 18 2012
Starting with offset 1, the sequence is the binomial transform of (1, 10, 9, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

T. D. Noe, Table of n, a(n) for n = 0..1000
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A007586.
Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.
Cf. A000217, A004188, A131432, A188892, A195160, A218470.

[n*(9*n-7)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 01 2015

Table[n (9n-7)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,11},51] (* Harvey P. Dale, May 07 2012 *)

a(n)=(9*n-7)*n/2 \\ Charles R Greathouse IV, Jun 16 2011

a(n) = n*(9*n-7)/2.
G.f.: x*(1+8*x)/(1-x)^3.
Row sums of triangle A131432. - Gary W. Adamson, Jul 10 2007
a(n) = 9*n + a(n-1) - 8 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=1, a(2)=11. - Harvey P. Dale, May 07 2012
a(n) = A218470(9n). - Philippe Deléham, Mar 27 2013
a(9*a(n)+37*n+1) = a(9*a(n)+37*n) + a(9*n+1). - Vladimir Shevelev, Jan 24 2014
a(n+y) - a(n-y-1) = (a(n+x) - a(n-x-1))*(2*y+1)/(2*x+1), 0 <= x < n, y <= x, a(0)=0. - Gionata Neri, May 03 2015
a(n) = A000217(n-1) + A000217(3*n-2) - A000217(n-2). - Charlie Marion, Dec 21 2019
Product_{n>=2} (1 - 1/a(n)) = 9/11. - Amiram Eldar, Jan 21 2021
E.g.f.: exp(x)*x*(2 + 9*x)/2. - Stefano Spezia, Dec 25 2022

A052224 Numbers whose sum of digits is 10.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613, 622, 631, 640

Offset: 1

Henry Bottomley, Feb 01 2000

Proper subsequence of A017173. - Rick L. Shepherd, Jan 12 2009
Subsequence of A227793. - Michel Marcus, Sep 23 2013
A007953(a(n)) = 10; number of repdigits = #{55,22222,1^10} = A242627(10) = 3. - Reinhard Zumkeller, Jul 17 2014
a(n) = A094677(n) for n = 1..28. - Reinhard Zumkeller, Nov 08 2015
The number of terms having <= m digits is the coefficient of x^10 in sum(i=0,9,x^i)^m = ((1-x^10)/(1-x))^m. - David A. Corneth, Jun 04 2016
In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 3921 terms from T. D. Noe)

Cf. A007953, A017173, A228915, A254524.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.
Cf. A094677.
Cf. A018900, A187813.
Sum of base-b digits equal b: A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9).

a052224 n = a052224_list !! (n-1)
a052224_list = filter ((== 10) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1000] | &+Intseq(n) eq 10 ]; // Vincenzo Librandi, Mar 10 2013

sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if sd(n) = 10 then n else end if end proc: seq(a(n), n = 1 .. 800); # Emeric Deutsch, Jan 16 2009

Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 7]], {s, Rest[IntegerPartitions[10]]}]]] (* T. D. Noe, Mar 08 2013 *)
Select[Range[1000], Total[IntegerDigits[#]] == 10 &] (* Vincenzo Librandi, Mar 10 2013 *)

isok(n) = sumdigits(n) == 10; \\ Michel Marcus, Dec 28 2015

\\ This algorithm needs a modified binomial.
C(n, k)=if(n>=k, binomial(n, k), 0)
\\ ways to roll s-q with q dice having sides 0 through n - 1.
b(s, q, n)=if(s<=q*(n-1), s+=q; sum(i=0, q-1, (-1)^i*C(q, i)*C(s-1-n*i, q-1)), 0)
\\ main algorithm; this program applies to all sequences of the form "Numbers whose sum of digits is m."
a(n,{m=10}) = {my(q); q = 2; while(b(m, q, 10) < n, q++); q--; s = m; os = m; r=0; while(q, if(b(s, q, 10) < n, n-=b(s, q, 10); s--, r+=(os-s)*10^(q); os = s; q--)); r+= s; r}
\\ David A. Corneth, Jun 05 2016

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits, b=10, sod=10): # works for any base, sum-of-digits
    alst = [sod] if 0 <= sod < b else []
    nzdigs = [i for i in range(1, b) if i <= sod]
    nzmultiset = []
    for d in range(1, b):
        nzmultiset += [d]*(sod//d)
    for d in range(2, maxdigits + 1):
        fullmultiset = [0]*(d-1-(sod-1)//(b-1)) + nzmultiset
        for firstdig in nzdigs:
            target_sum, restmultiset = sod - int(firstdig), fullmultiset[:]
            restmultiset.remove(firstdig)
            for p in multiset_permutations(restmultiset, d-1):
              if sum(p) == target_sum:
                  alst.append(int("".join(map(str, [firstdig]+p)), b))
                  if p[0] == target_sum:
                      break
    return alst
print(auptodigs(4)) # Michael S. Branicky, Sep 14 2021

def A052224(N = 19):
    """Return a generator of the sequence of all integers >= N with the same
    digit sum as N."""
    while True:
        yield N
        N = A228915(N) # skip to next larger integer with the same digit sum
a = A052224(); [next(a) for  in range(50)] # _M. F. Hasler, Mar 16 2022

a(n+1) = A228915(a(n)) for any n > 0. - Rémy Sigrist, Jul 10 2018

Incorrect formula deleted by N. J. A. Sloane, Jan 15 2009
Extended by Emeric Deutsch, Jan 16 2009
Offset changed by Bruno Berselli, Mar 07 2013

A056991 Numbers with digital root 1, 4, 7 or 9.

Original entry on oeis.org

1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142

Offset: 1

Eric W. Weisstein

All squares are members (see A070433).
May also be defined as: possible sums of digits of squares. - Zak Seidov, Feb 11 2008
First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - Zak Seidov, Feb 11 2008
Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.
a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - Gary Detlefs, Mar 19 2010
From Klaus Purath, Feb 20 2023: (Start)
The sum of digits of any term belongs to the sequence. Also the products of any terms belong to the sequence.
This is the union of A017173, A017209, A017245 and A008591.
Positive integers of the forms x^2 + (2*m+1)*x*y + (m^2+m-2)*y^2, for integers m.
This sequence is closed under multiplication. (End)

R. J. Mathar, Table of n, a(n) for n = 1..22222
H. I. Okagbue, M. O. Adamu, S. A. Iyase, and A. A. Opanuga, Sequence of Integers Generated by Summing the Digits of their Squares, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
Eric Weisstein's World of Mathematics, Square Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000290, A008591, A017173, A017209, A017245, A056992, A070433.

For complement see A268226.

seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # Gary Detlefs, Mar 19 2010

LinearRecurrence[{1,0,0,1,-1},{1,4,7,9,10},70] (* Harvey P. Dale, Aug 29 2015 *)

forstep(n=1,1e3,[3,3,2,1],print1(n", ")) \\ Charles R Greathouse IV, Sep 21 2012

From R. J. Mathar, Feb 14 2008: (Start)
O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).
a(n) = a(n-4) + 9. (End)
a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - Gary Detlefs, Mar 19 2010
a(n) = a(n-1)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 27 2021
a(n) = 3*n - floor(n/4) - 2*floor((n+3)/4). - Ridouane Oudra, Jan 21 2024
E.g.f.: (cos(x) + (9*x - 1)*cosh(x) - 3*sin(x) + (9*x - 2)*sinh(x))/4. - Stefano Spezia, Feb 21 2024

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A161705 a(n) = 18*n + 1.

Original entry on oeis.org

1, 19, 37, 55, 73, 91, 109, 127, 145, 163, 181, 199, 217, 235, 253, 271, 289, 307, 325, 343, 361, 379, 397, 415, 433, 451, 469, 487, 505, 523, 541, 559, 577, 595, 613, 631, 649, 667, 685, 703, 721, 739, 757, 775, 793, 811, 829, 847, 865, 883, 901, 919, 937, 955

Offset: 0

Reinhard Zumkeller, Jun 17 2009

Digital root of a(n) is 1. - Alexander R. Povolotsky, Jun 13 2012

These numbers can be written as the sum of four integer cubes as a(n) = (2*n + 14)^3 + (3*n + 30)^3 + (- 2*n - 23)^3 + (- 3*n - 26)^3. - Arkadiusz Wesolowski, Aug 15 2013

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161709, A161714, A287326 (fourth column).

Cf. A016777, A017173.

List([0..60], n-> 18*n+1); # G. C. Greubel, Sep 18 2019

[18*n +1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(18*n+1, n=0..60); # G. C. Greubel, Sep 18 2019

Range[1, 1000, 18] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
LinearRecurrence[{2,-1},{1,19}, 60] (* G. C. Greubel, Feb 17 2017 *)

vector(60, n, 18*n-17) \\ G. C. Greubel, Feb 17 2017

[18*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 18*n + 1, n >= 0.
a(n) = a(n-1) + 18 (with a(0)=1). - Vincenzo Librandi, Dec 27 2010
From G. C. Greubel, Feb 17 2017: (Start)
G.f.: (1 + 17*x)/(1-x)^2.
E.g.f.: (1 + 18*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)
a(n) = A017173(2*n) = A016777(6*n). - Elmo R. Oliveira, Apr 12 2025

A017185 a(n) = 9*n + 2.

Original entry on oeis.org

2, 11, 20, 29, 38, 47, 56, 65, 74, 83, 92, 101, 110, 119, 128, 137, 146, 155, 164, 173, 182, 191, 200, 209, 218, 227, 236, 245, 254, 263, 272, 281, 290, 299, 308, 317, 326, 335, 344, 353, 362, 371, 380, 389, 398, 407, 416, 425, 434, 443, 452, 461, 470, 479

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 2. - Cino Hilliard, Dec 26 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A016789, A017173.

[9*n+2: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[2, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{2,11},60] (* Harvey P. Dale, Jun 09 2023 *)

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

G.f.: (2 + 7*x)/(x - 1)^2. - R. J. Mathar, Jul 14 2016
E.g.f.: exp(x)*(2 + 9*x). - Stefano Spezia, Dec 07 2024
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A016789(3*n). (End)

More terms from Cino Hilliard, Dec 26 2006

A093644 (9,1) Pascal triangle.

Original entry on oeis.org

1, 9, 1, 9, 10, 1, 9, 19, 11, 1, 9, 28, 30, 12, 1, 9, 37, 58, 42, 13, 1, 9, 46, 95, 100, 55, 14, 1, 9, 55, 141, 195, 155, 69, 15, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 9, 73, 260, 532, 686, 574, 308, 100, 17, 1, 9, 82, 333, 792, 1218, 1260, 882, 408, 117, 18, 1, 9, 91, 415

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(9;n,m) gives in the columns m>=1 the figurate numbers based on A017173, including the 11-gonal numbers A051682 (see the W. Lang link).
This is the ninth member, d=9, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5, for d=1..8.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+8*z)/(1-(1+x)*z).
The SW-NE diagonals give A022099(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 8. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011

			Triangle begins
  [1];
  [9,  1];
  [9, 10,  1];
  [9, 19, 11,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Wolfdieter Lang, First 10 rows and array of figurate numbers .

Row sums: A020714(n-1), n >= 1, 1 for n=0, alternating row sums are 1 for n=0, 8 for n=2 and 0 otherwise.
The column sequences give for m=1..9: A017173, A051682 (11-gonal), A007586, A051798, A051879, A050405, A052206, A056117, A056003.
Cf. A093645 (d=10).

a093644 n k = a093644_tabl !! n !! k
a093644_row n = a093644_tabl !! n
a093644_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [9, 1]
-- Reinhard Zumkeller, Aug 31 2014

Join[{1},Table[Binomial[n,k]+8Binomial[n-1,k],{n,20},{k,0,n}]//Flatten] (* Harvey P. Dale, Aug 17 2024 *)

a(n, m) = F(9;n-m, m) for 0 <= m <= n, otherwise 0, with F(9;0, 0)=1, F(9;n, 0)=9 if n >= 1 and F(9;n, m):=(9*n+m)*binomial(n+m-1, m-1)/m if m >= 1.
Recursion: a(n, m)=0 if m > n, a(0, 0)= 1; a(n, 0)=9 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+8*x)/(1-x)^(m+1), m >= 0.
T(n, k) = C(n, k) + 8*C(n-1, k). - Philippe Deléham, Aug 28 2005
Row n: Expansion of (9+x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 10 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(9 + 19*x + 11*x^2/2! + x^3/3!) = 9 + 28*x + 58*x^2/2! + 100*x^3/3! + 155*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
G.f.: (-1-8*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015

A116371 Number of partitions of n into parts with digital root = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 67, 68

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(18) = #{10+8x1, 18x1} = 2;
a(19) = #{19, 10+9x1, 19x1} = 3;
a(20) = #{19+1, 10+10, 10+10x1, 19x1} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
A017173, A156144, A156145. [From Reinhard Zumkeller, Feb 05 2009]

a116371 n = p a017173_list 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

A143683 Pascal-(1,8,1) array.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 10,   19,    28,     37,      46,       55, ... A017173;
  1, 19,  118,   298,    559,     901,     1324, ...
  1, 28,  298,  1540,   4483,    9856,    18388, ...
  1, 37,  559,  4483,  21286,   67006,   164242, ...
  1, 46,  901,  9856,  67006,  304300,  1004590, ...
  1, 55, 1324, 18388, 164242, 1004590,  4443580, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 10,   1;
  1, 19,  19,    1;
  1, 28, 118,   28,    1;
  1, 37, 298,  298,   37,   1;
  1, 46, 559, 1540,  559,  46,  1;
  1, 55, 901, 4483, 4483, 901, 55, 1;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf.Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7).

Cf. A003683, A017173, A143680.

a143683 n k = a143683_tabl !! n !! k
a143683_row n = a143683_tabl !! n
a143683_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021

cp's OEIS Frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

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A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

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A051682 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.

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A052224 Numbers whose sum of digits is 10.

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A056991 Numbers with digital root 1, 4, 7 or 9.

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A051682 11-gonal (or hendecagonal) numbers: a(n) = n(9n-7)/2.