A017281 -id:A017281 - cp's OEIS frontend

A143685 Pascal-(1,9,1) array.

1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 11,   21,    31,     41,      51,       61, ... A017281;
  1, 21,  141,   361,    681,    1101,     1621, ...
  1, 31,  361,  1991,   5921,   13151,    24681, ...
  1, 41,  681,  5921,  29761,   96201,   239241, ...
  1, 51, 1101, 13151,  96201,  460251,  1565301, ...
  1, 61, 1621, 24681, 239241, 1565301,  7272861, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 11,    1;
  1, 21,   21,     1;
  1, 31,  141,    31,     1;
  1, 41,  361,   361,    41,     1;
  1, 51,  681,  1991,   681,    51,    1;
  1, 61, 1101,  5921,  5921,  1101,   61,  1;
  1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A002534, A143680, A143682.

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), A143683 (m = 8), this sequence (m = 9).

A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Square array: T(n, k) = T(n, k-1) + 9*T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(0, k) = 1.
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.
Riordan array (1/(1-x), x*(1+9*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - Jean-François Alcover, May 24 2013
Sum_{k=0..n} T(n, k) = A002534(n+1). - G. C. Greubel, May 29 2021

A159551 a(n) = 101*n + 10.

Original entry on oeis.org

10, 111, 212, 313, 414, 515, 616, 717, 818, 919, 1020, 1121, 1222, 1323, 1424, 1525, 1626, 1727, 1828, 1929, 2030, 2131, 2232, 2333, 2434, 2535, 2636, 2737, 2838, 2939, 3040, 3141, 3242, 3343, 3444, 3545, 3646, 3747, 3848, 3949, 4050, 4151, 4252, 4353, 4454, 4555

Offset: 0

Robert G. Wilson v, Apr 14 2009

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

Cf. A017281, A017509.

[101*n+10: n in [0..50]]; // Vincenzo Librandi, Jul 30 2011

f[n_] := FromDigits[ IntegerDigits[n^3 + n^2 + n - 1, n + 1]]; Array[f, 54]

From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: (10 + 91*x)/(1-x)^2.
E.g.f.: exp(x)*(10 + 101*x).
a(n) = 2*a(n-1) - a(n-2). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Jul 30 2011

A187715 a(n) = 5*n - (9 + (-1)^n)/2.

Original entry on oeis.org

1, 5, 11, 15, 21, 25, 31, 35, 41, 45, 51, 55, 61, 65, 71, 75, 81, 85, 91, 95, 101, 105, 111, 115, 121, 125, 131, 135, 141, 145, 151, 155, 161, 165, 171, 175, 181, 185, 191, 195, 201, 205, 211, 215, 221, 225

Offset: 1

Vincenzo Librandi, Mar 13 2011

Numbers congruent to {1,5} mod 10. - Bruno Berselli, Mar 31 2012

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

Cf. A001622, A010711 (first differences), A017281, A017329.

Filtered([1..250],n-> n mod 10 =1 or n mod 10 = 5); # Muniru A Asiru, Nov 25 2018

Magma
```
[5*n -(9+(-1)^n)/2: n in [1..60]];
```

[5*n-(9+(-1)^n)/2$n=1..50]; # Muniru A Asiru, Nov 25 2018

nxt[{n_,a_}]:={n+1,If[EvenQ[n+1],a+4,a+6]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* Harvey P. Dale, Feb 16 2013 *)
Table[BitOr[5*n, 1], {n, 0, 50}] (* Jon Maiga, Nov 24 2018 *)

vector(50, n, (10*n -9-(-1)^n)/2) \\ G. C. Greubel, Dec 04 2018

for n in range(1,60): print(int(5*n - (9 + (-1)**n)/2), end=', ') # Stefano Spezia, Nov 30 2018

[(10*n -9-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Dec 04 2018

a(n) = a(n-1) + 4 if n is even, a(n) = a(n-1) + 6 if n is odd.
a(n) = 2*a(n-1) - a(n-2) - 2*(-1)^n.
From R. J. Mathar, Mar 15 2011: (Start)
G.f.: x*(1 + 4*x + 5*x^2)/( (1+x)*(1-x)^2 ).
Bisections: a(2*n+1) = A017281(n), a(2*n) = A017329(n-1). (End)
a(n) = 5*(n-1) bitwise-OR 1. - Jon Maiga, Nov 24 2018
E.g.f.: ((10*x-9)*exp(x) - exp(-x) + 10)/2. - G. C. Greubel, Dec 04 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5+2*sqrt(5))*Pi/20 + 3*log(phi)/(4*sqrt(5)) + log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

Definition rewritten by R. J. Mathar, Mar 15 2011

A216405 Numbers which start a run of nine consecutive zero-digit-free decimal integers, each of which is divisible by the sum of its digits.

Original entry on oeis.org

1, 142813628717821, 253323932621811, 1234954171531131, 1713763544613181, 3713154346661821, 5953112416611411, 8711631351783421, 11853531183574141, 12191214257422251, 17137635446131261, 19941476493818971, 21342541323383331, 25628491758925521, 28665872459864731

Offset: 1

Jack Brennen, Oct 16 2012

Each term of the sequence ends with the digit 1.

No run of ten consecutive zero-digit-free decimal integers is possible.

			The numbers from a(2)=142813628717821 to 142813628717829 are each divisible by their digit sums, which are 61 to 69 respectively.

Subsequence of A217973 and of A017281.

\\ Algorithm from Jack Brennen
list(lim)=my(v=List([1]),m); forstep(d=11, (40320*lim)^(1/9), 10, m=lcm(vector(9,k,d+k-1)); forstep(x=m+d, lim, m, if(sumdigits(x)==d && vecsort(digits(x))[1], listput(v,x)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 16 2012

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset: 0

Kival Ngaokrajang, Jul 07 2014

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
The triangle under the main diagonal is A121722.
Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

			Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

A266297 Numbers whose last digit is a square.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Wesley Ivan Hurt, Dec 26 2015

Numbers ending in 0, 1, 4 and 9.
Union of A008592, A017281, A017317 and A017377. - Hurt
None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015
Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015
Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

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

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

G.f.: x^2*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)).
a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.
a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.
a(n+1) - a(n) = A091084(n+1) for n>0.
Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024

A271508 Numbers that are congruent to {1,4} mod 10.

Original entry on oeis.org

1, 4, 11, 14, 21, 24, 31, 34, 41, 44, 51, 54, 61, 64, 71, 74, 81, 84, 91, 94, 101, 104, 111, 114, 121, 124, 131, 134, 141, 144, 151, 154, 161, 164, 171, 174, 181, 184, 191, 194, 201, 204, 211, 214, 221, 224, 231, 234, 241, 244, 251, 254, 261, 264, 271, 274

Offset: 1

Wesley Ivan Hurt, Apr 08 2016

Numbers ending in 1 or 4, Union of A017281 and A017317.

a(n+3) gives the sum of 5 consecutive terms of A004442 starting at A004442(n) for n>0. (i.e., a(4) = 14 = 0+3+2+5+4 = Sum_{i=0..4} A004442(n+i)).

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A004442, A017281, A017317, A042948, A047241.

Magma
```
[5*n-5-(-1)^n : n in [1..100]];
```

A271508:=n->5*n-5-(-1)^n: seq(A271508(n), n=1..100);

Table[5 n - 5 - (-1)^n, {n, 60}] (* or *)
Select[Range[0, 300], MemberQ[{1, 4}, Mod[#, 10]] &]

my(x='x+O('x^99)); Vec(x*(1+3*x+6*x^2)/((-1+x)^2*(1+x))) \\ Altug Alkan, Apr 09 2016

G.f.: x*(1+3*x+6*x^2)/((-1+x)^2*(1+x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 5*n - 5 - (-1)^n.
a(n) = -n + 2*A047241(n).
a(n+1) = n + 1 + 2*A042948(n).
Shifted bisections: a(2n+2) = A017317(n), a(2n+1) = A017281(n).
E.g.f.: 5*(x-1)*exp(x) - exp(-x). - G. C. Greubel, Apr 08 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

A323178 a(n) = 1 + 100*n^2 for n >= 0.

Original entry on oeis.org

1, 101, 401, 901, 1601, 2501, 3601, 4901, 6401, 8101, 10001, 12101, 14401, 16901, 19601, 22501, 25601, 28901, 32401, 36101, 40001, 44101, 48401, 52901, 57601, 62501, 67601, 72901, 78401, 84101, 90001, 96101, 102401, 108901

Offset: 0

Paul Curtz, Jan 06 2019

nonn

Terms of A261327 ending in 1 (01 for n > 0.)
a(n) mod 9 = period 9: repeat [1, 2, 5, 1, 8, 8, 1, 5, 2] = A275704(n+3).
(Analogous sequence: b(n) = 29 + 100*n*(n+1) = A261327(A017329) = 29, 229, 629, ... .)

Cf. A000290, A008602 (20*n), A261327, A275704.

Subsequence of A017281.

a[n_] := 1 + 100*n^2 ; Array[a, 50, 0] (* or *)
CoefficientList[Series[(-1 - 98 x - 101 x^2)/(-1 + x)^3, {x, 0, 50}], x] (* or *)
CoefficientList[Series[E^x (1 + 100 x + 100 x^2), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Jan 06 2019 *)

a(n) = A261327(A008602(n)).
Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2 with initial values a(0) = 1, a(1) = 101 and a(2) = 401.
From Stefano Spezia, Jan 06 2019: (Start)
O.g.f.: (-1 - 98*x - 101*x^2)/(-1 + x)^3.
E.g.f.: exp(x)*(1 + 100*x + 100*x^2).
(End)

Corrected and extended (recurrence formula) by Werner Schulte, Feb 18 2019

cp's OEIS Frontend

A143685 Pascal-(1,9,1) array.

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A159551 a(n) = 101*n + 10.

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A187715 a(n) = 5*n - (9 + (-1)^n)/2.

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A216405 Numbers which start a run of nine consecutive zero-digit-free decimal integers, each of which is divisible by the sum of its digits.

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PARI

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

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A266297 Numbers whose last digit is a square.

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Magma

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Mathematica

PARI

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A271508 Numbers that are congruent to {1,4} mod 10.

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