A156145 -id:A156145 - 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

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

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

cp's OEIS Frontend

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

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

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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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Haskell