A017293 -id:A017293 - cp's OEIS frontend

A139222 a(n) = 30*n - 27.

3, 33, 63, 93, 123, 153, 183, 213, 243, 273, 303, 333, 363, 393, 423, 453, 483, 513, 543, 573, 603, 633, 663, 693, 723, 753, 783, 813, 843, 873, 903, 933, 963, 993, 1023, 1053, 1083, 1113, 1143, 1173, 1203, 1233, 1263, 1293, 1323, 1353, 1383, 1413, 1443, 1473

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 3 with the units digit equal to 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139245, A017329, A139249, A139264, A139279 and A139280.

[30*n - 27: n in [1..60]]; // Vincenzo Librandi, Jun 17 2011

Range[3, 2000, 30] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
30*Range[50]-27 (* or *) NestList[30+#&,3,50] (* Harvey P. Dale, Mar 27 2021 *)

a(n)=30*n-27 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 30.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 3*x*(1+9*x)/(1-x)^2.
E.g.f.: 3*(exp(x)*(10*x - 9) + 9).
a(n) = 3*A017281(n-1) = A139280(n)/3.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A139249 a(n) = 30*n - 24.

Original entry on oeis.org

6, 36, 66, 96, 126, 156, 186, 216, 246, 276, 306, 336, 366, 396, 426, 456, 486, 516, 546, 576, 606, 636, 666, 696, 726, 756, 786, 816, 846, 876, 906, 936, 966, 996, 1026, 1056, 1086, 1116, 1146, 1176, 1206, 1236, 1266, 1296, 1326, 1356, 1386, 1416, 1446, 1476

Offset: 1

Odimar Fabeny, Jun 06 2008, Jun 07 2008

Multiples of 6 with unit digit equal to 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139264, A139279 and A139280. - Reinhard Zumkeller, Jun 22 2008

Cf. A016861.

[30*n - 24: n in [1..60]]; // Vincenzo Librandi, Jun 19 2011

Range[6,7000,30] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2011 *)
NestList[30+#&,6,50] (* or *) LinearRecurrence[{2,-1},{6,36},50] (* Harvey P. Dale, May 27 2018 *)

a(n)=30*n-24 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 30.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 6*x*(1+4*x)/(1-x)^2.
E.g.f.: 6*(exp(x)*(5*x - 4) + 4).
a(n) = 6*A016861(n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

Edited by R. J. Mathar, Jul 20 2008

A139264 a(n) = 70*n - 63.

Original entry on oeis.org

7, 77, 147, 217, 287, 357, 427, 497, 567, 637, 707, 777, 847, 917, 987, 1057, 1127, 1197, 1267, 1337, 1407, 1477, 1547, 1617, 1687, 1757, 1827, 1897, 1967, 2037, 2107, 2177, 2247, 2317, 2387, 2457, 2527, 2597, 2667, 2737, 2807, 2877, 2947, 3017, 3087, 3157, 3227

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 7 with unit digit equal to 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139279 and A139280.

[70*n-63: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

70*Range[50]-63 (* Harvey P. Dale, May 06 2019 *)

a(n)=70*n-63 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 70.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 7*x*(1+9*x)/(1-x)^2.
E.g.f.: 7*(exp(x)*(10*x - 9) + 9).
a(n) = 7*A017281(n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A166726 Nonnegative integers with English names ending in "o".

Original entry on oeis.org

0, 2, 22, 32, 42, 52, 62, 72, 82, 92, 102, 122, 132, 142, 152, 162, 172, 182, 192, 202, 222, 232, 242, 252, 262, 272, 282, 292, 302, 322, 332, 342, 352, 362, 372, 382, 392, 402, 422, 432, 442, 452, 462, 472, 482, 492, 502, 522, 532, 542, 552, 562, 572, 582

Offset: 1

Rick L. Shepherd, Oct 20 2009

Two (2) is the only prime number whose English name does not end in "e" or "n".

			Zero (0) is a term; thirty-two (32) is a term; twelve (12) is not a term (but is a term of A059093).

Cf. A017293, A166727, A166728, A166729, A166730, A166731, A059093, A060228.

{0} UNION (A017293 MINUS {n | n = 12 mod 100}).

A260181 Numbers whose last digit is prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147

Offset: 1

Wesley Ivan Hurt, Jul 17 2015

Numbers ending in 2, 3, 5 or 7.
The subsequence of primes is A042993. - Michel Marcus, Jul 19 2015
From Wesley Ivan Hurt, Aug 15 2015, Sep 26 2015: (Start)
Ceiling(a(n)/2) = A047201(n).
Complement of (A197652 Union A262389). (End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A042993, A047201, A092620, subset of A118950.
Union of A017293, A017305, A017329 and A017353.
First differences are [1,2,2,5,...] = A002522(A140081(n-1)).
Cf. A197652, A262389.

a:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2; List([1..60],n->a(n)); # Muniru A Asiru, Feb 16 2018

[(5*n-4-(-1)^n+((3-(-1)^n) div 2)*(-1)^((2*n+5-(-1)^n) div 4))/2: n in [1..70]]; // Vincenzo Librandi, Jul 18 2015

A260181:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2: seq(A260181(n), n=1..100);

CoefficientList[Series[(2 + x + 2 x^2 + 2 x^3 + 3 x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 5, 7, 12}, 60] (* Vincenzo Librandi, Jul 18 2015 *)
Table[(5n - 4 - (-1)^n + ((3 - (-1)^n)/2)*(-1)^((2*n + 5 - (-1)^n)/4))/2, {n, 100}] (* Wesley Ivan Hurt, Aug 11 2015 *)

is(n)=my(m=digits(n));isprime(m[#m]) \\ Anders Hellström, Jul 19 2015

A260181(n)=(n--)\4*10+prime(n%4+1) \\ is(n)=isprime(n%10) is much more efficient than the above. - M. F. Hasler, Sep 16 2016

G.f.: x*(2+x+2*x^2+2*x^3+3*x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1)+a(n-4)-a(n-5), n>5.
a(n) = (5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(5*sqrt(5+2*sqrt(5))) - 25*log(5) - 40*log(2) + 5*sqrt(5)*arccoth(843/2))/200. - Amiram Eldar, Jul 30 2024

A168457 a(n) = (10n + 5(-1)^n - 1)/2.

Original entry on oeis.org

2, 12, 12, 22, 22, 32, 32, 42, 42, 52, 52, 62, 62, 72, 72, 82, 82, 92, 92, 102, 102, 112, 112, 122, 122, 132, 132, 142, 142, 152, 152, 162, 162, 172, 172, 182, 182, 192, 192, 202, 202, 212, 212, 222, 222, 232, 232, 242, 242, 252, 252, 262, 262, 272, 272, 282

Offset: 1

Vincenzo Librandi, Nov 26 2009

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

Cf. A017293, A168459, A168282.

[2+10*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[5 n + 5 (-1)^n/2 - 1/2, {n, 60}] (* Bruno Berselli, Sep 16 2013 *)
Table[2 + 10 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[2 (1 + 5 x - x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{2,12,12},80] (* Harvey P. Dale, Dec 27 2024 *)

a(n) = 10*n - a(n-1) - 6 for n>1, a(1)=2.
From Bruno Berselli, Sep 16 2013: (Start)
G.f.: 2*x*(1 + 5*x - x^2)/((1+x)*(1-x)^2).
a(n) = A168459(n) + 1 = 2*A168282(n).
a(n) = a(n-1) +a(n-2) -a(n-3). (End)
a(n) = 2 + 10*Floor(n/2). -  Vincenzo Librandi, Sep 19 2013
E.g.f.: (1/2)*(5 - 4*exp(x) + (10*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

New definition by Bruno Berselli, Sep 16 2013

A017294 a(n) = (10*n+2)^2.

Original entry on oeis.org

4, 144, 484, 1024, 1764, 2704, 3844, 5184, 6724, 8464, 10404, 12544, 14884, 17424, 20164, 23104, 26244, 29584, 33124, 36864, 40804, 44944, 49284, 53824, 58564, 63504, 68644, 73984, 79524, 85264

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016861, A017293.

[(10*n+2)^2: n in [0..35]]; // Vincenzo Librandi, Jul 30 2011

(10 Range[0, 50] + 2)^2 (* Wesley Ivan Hurt, Sep 04 2022 *)

a(n)=(10*n+2)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1)+200n-60, n>0 ; a(0)=4. - Miquel Cerda, Oct 30 2016
G.f.: 4*(1 + 33*x + 16*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Oct 30 2016
a(n) = A017293(n)^2. - Michel Marcus, Oct 30 2016
a(n) = (2*A016861(n))^2. - David A. Corneth, Oct 30 2016

A346629 Number of n-digit positive integers that are the product of two integers ending with 2.

Original entry on oeis.org

1, 4, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Stefano Spezia, Jul 25 2021

a(n) is the number of n-digit numbers in A139245.

After initial 1 or 2 values the same as A137233. - R. J. Mathar, Aug 23 2021

Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to digits.

Cf. A011557 (powers of 10), A017293 (positive integers ending with 2), A052268 (number of n-digit integers), A139245 (product of two integers ending with 2), A093143, A337855, A337856.

Cf. A137233.

Mathematica
```
LinearRecurrence[{10},{1,4,45},25]
```

O.g.f.: x*(1 - 6*x + 5*x^2)/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 + 110*x - 50*x^2)/200.
a(n) = 10*a(n-1) for n > 3, with a(1) = 1, a(2) = 4 and a(3) = 45.
a(n) = 45*10^(n-3) for n > 2.
a(n) = 45*A011557(n-3) for n > 2.
Sum_{i=1..n} a(n) = A093143(n-1).

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A017295 (10*n+2)^3.

Original entry on oeis.org

8, 1728, 10648, 32768, 74088, 140608, 238328, 373248, 551368, 778688, 1061208, 1404928, 1815848, 2299968, 2863288, 3511808, 4251528, 5088448, 6028568, 7077888, 8242408, 9528128, 10941048, 12487168

Offset: 0

N. J. A. Sloane, Dec 11 1996

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A017293.

[(10*n+2)^3: n in [0..35]]; // Vincenzo Librandi, Jul 30 2011

A017295:=n->(10*n+2)^3; seq(A017295(n), n=0..40); # Wesley Ivan Hurt, May 02 2014

Table[(10 n + 2)^3, {n, 0, 40}] (* Wesley Ivan Hurt, May 02 2014 *)
LinearRecurrence[{4,-6,4,-1},{8,1728,10648,32768},30] (* Harvey P. Dale, Aug 24 2018 *)

Vec(8*(64*x^3+473*x^2+212*x+1)/(x-1)^4 + O(x^100)) \\ Colin Barker, May 03 2014

a(n) = A017293(n)^3 = A000578(10n+2). - Wesley Ivan Hurt, May 02 2014

G.f.: 8*(64*x^3+473*x^2+212*x+1) / (x-1)^4. - Colin Barker, May 03 2014

cp's OEIS Frontend

A139222 a(n) = 30*n - 27.

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A139249 a(n) = 30*n - 24.

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A139264 a(n) = 70*n - 63.

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A166726 Nonnegative integers with English names ending in "o".

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A260181 Numbers whose last digit is prime.

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A168457 a(n) = (10*n + 5*(-1)^n - 1)/2.

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A017294 a(n) = (10*n+2)^2.

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A168457 a(n) = (10n + 5(-1)^n - 1)/2.