A017629 -id:A017629 - cp's OEIS frontend

A096025 Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 6 and (n+7) mod 9 <> 1.

843, 1683, 3363, 4203, 5883, 6723, 8403, 9243, 10923, 11763, 13443, 14283, 15963, 16803, 18483, 19323, 21003, 21843, 23523, 24363, 26043, 26883, 28563, 29403, 31083, 31923, 33603, 34443, 36123, 36963, 38643, 39483, 41163, 42003, 43683

Offset: 1

Klaus Brockhaus, Jun 15 2004

Numbers n such that n mod 840 = 3 and n mod 2520 <> 3.

			843 mod 2 = 844 mod 3 = 845 mod 4 = 846 mod 5 = 847 mod 6 = 848 mod 7 = 849 mod 8 = 1 and 850 mod 9 = 4, hence 843 is in the sequence.

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

Cf. A007310, A017629, A096022, A096023, A096024, A096026, A096027.

[n: n in [1..44000] | forall{j: j in [0..6] | IsOne((n+j) mod (2+j)) and (n+7) mod 9 ne 1}]; // Bruno Berselli, Apr 11 2013

LinearRecurrence[{1,1,-1},{843,1683,3363},40] (* Harvey P. Dale, Nov 22 2015 *)

{k=7;m=44000;for(n=1,m,j=0;b=1;while(b&&j

a(n) = -3*(209+70*(-1)^n-420*n). a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: 3*x*(279*x^2+280*x+281) / ((x-1)^2*(x+1)). - Colin Barker, Apr 11 2013

A096027 Numbers k such that (k+j) mod (2+j) = 1 for j from 0 to 10 and (k+11) mod 13 <> 1.

Original entry on oeis.org

27723, 55443, 83163, 110883, 138603, 166323, 194043, 221763, 249483, 277203, 304923, 332643, 388083, 415803, 443523, 471243, 498963, 526683, 554403, 582123, 609843, 637563, 665283, 693003, 748443, 776163, 803883, 831603, 859323, 887043

Offset: 1

Klaus Brockhaus, Jun 15 2004

Numbers k such that k mod 27720 = 3 and k mod 360360 <> 3.

			27723 mod 2 = 27724 mod 3 = 27725 mod 4 = 27726 mod 5 = 27727 mod 6 = 27728 mod 7 = 27729 mod 8 = 27730 mod 9 = 27731 mod 10 = 27731 mod 11 = 27731 mod 12 = 1 and 27732 mod 13 = 3, hence 27723 is in the sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A007310, A017629, A096022, A096023, A096024, A096025, A096026.

[n: n in [1..900000] | forall{j: j in [0..10] | IsOne((n+j) mod (2+j)) and (n+11) mod 13 ne 1}]; // Bruno Berselli, Apr 11 2013

{k=11;m=900000;for(n=1,m,j=0;b=1;while(b&&j

G.f.: 3*x*(9239*x^12 +9240*x^11 +9240*x^10 +9240*x^9 +9240*x^8 +9240*x^7 +9240*x^6 +9240*x^5 +9240*x^4 +9240*x^3 +9240*x^2 +9240*x +9241) / ((x -1)^2*(x +1)*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)*(x^4 -x^2 +1)). - Colin Barker, Apr 11 2013

A195319 Three times second hexagonal numbers: 3n(2*n+1).

Original entry on oeis.org

0, 9, 30, 63, 108, 165, 234, 315, 408, 513, 630, 759, 900, 1053, 1218, 1395, 1584, 1785, 1998, 2223, 2460, 2709, 2970, 3243, 3528, 3825, 4134, 4455, 4788, 5133, 5490, 5859, 6240, 6633, 7038, 7455, 7884, 8325, 8778, 9243, 9720, 10209, 10710, 11223

Offset: 0

Omar E. Pol, Sep 17 2011

Sequence found by reading the line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Semi-axis opposite to A094159 in the same spiral.
Sum of the numbers from 2*n to 4*n. - Wesley Ivan Hurt, Nov 27 2015
From Peter M. Chema, Jan 21 2017: (Start)
Also 0 together with the partial sums of A017629.
Digit root is 0 together with period 3: repeat [9,3,9].
Final digits cycle a length period 10: repeat [0,9,0,3,8,5,4,5,8,3]. (End)
Sequence found by reading the line from 0, in the direction 0, 9, ..., in the triangle spiral. - Hans G. Oberlack, Dec 08 2018

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

Bisection of A045943.

Cf. A000217, A001318, A014105, A094159, A017629.

List([0..30], n -> 3*n*(2*n+1)); # G. C. Greubel, Dec 07 2018

[3*n*(2*n+1): n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

A195319:=n->6*n^2 + 3*n: seq(A195319(n), n=0..50); # Wesley Ivan Hurt, Nov 27 2015

Table[6n^2+3n,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,9,30},50] (* Harvey P. Dale, Oct 13 2013 *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 27 2015 *)

a(n)=3*n*(2*n+1) \\ Charles R Greathouse IV, Oct 16 2015

[3*n*(2*n+1) for n in range(50)] # G. C. Greubel, Dec 07 2018

a(n) = 6*n^2 + 3*n = 3*A014105(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, Oct 13 2013
G.f.: 3*x*(3+x) / (1-x)^3. - Wesley Ivan Hurt, Nov 27 2015
a(n) = A000217(3*n) + 3*A000217(n). - Bruno Berselli, Aug 31 2017
E.g.f.: 3*x*(2*x+3)*exp(x). - G. C. Greubel, Dec 07 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(1 - log(2))/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/2 + log(2) - 2)/3. (End)

A127989 a(n) = 2n^3 - 2n + 9.

Original entry on oeis.org

9, 21, 57, 129, 249, 429, 681, 1017, 1449, 1989, 2649, 3441, 4377, 5469, 6729, 8169, 9801, 11637, 13689, 15969, 18489, 21261, 24297, 27609, 31209, 35109, 39321, 43857, 48729, 53949, 59529, 65481, 71817, 78549, 85689, 93249, 101241, 109677, 118569

Offset: 1

Artur Jasinski, Feb 10 2007

All numbers generated by the irreducible polynomial 2*n^3-2*n+9 are composite (indeed, they are multiples of 3). Largest prime factor A127990, smallest prime factor A127991, number of prime factors A127992.

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

Cf. A127990, A127991, A127992. Subsequence of A017629.

[2*(n-1)*n*(n+1)+9: n in [1..50]]; // Vincenzo Librandi, Sep 13 2011

Table[2x^3 - 2x + 9, {x, 1, 100}]
LinearRecurrence[{4,-6,4,-1}, {9, 21, 57, 129}, 50] (* G. C. Greubel, May 08 2018 *)

a(n)=2*n^3-2*n+9 \\ Charles R Greathouse IV, Sep 30 2015

a(n) = 2*(n-1)*n*(n+1)+9 = 3*(2*A007290(n+1)+3).
G.f.: 3*x*(3-5*x+9*x^2-3*x^3)/(1-x)^4. - Bruno Berselli, Sep 09 2011
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Oct 11 2021

A017638 a(n) = (12n+9)^10.

Original entry on oeis.org

3486784401, 16679880978201, 1531578985264449, 34050628916015625, 362033331456891249, 2446194060654759801, 12157665459056928801, 48398230717929318249, 162889462677744140625, 480682838924478847449

Offset: 0

N. J. A. Sloane

From Fermat's little theorem, it follows that all terms are congruent to 1 mod 11 except when n is congruent to 2 mod 11 (because for those n, 12*n+9 is a multiple of 11). - Alonso del Arte, Dec 02 2013

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A011557, A017629.

[(12*n+9)^10 : n in [0..20]]; // Wesley Ivan Hurt, Nov 25 2021

A017638:=n->(12*n+9)^10; seq(A017638(n), n=0..20); # Wesley Ivan Hurt, Dec 02 2013

Table[(12n + 9)^10, {n, 0, 20}] (* Wesley Ivan Hurt, Dec 02 2013 *)

a(n) = (12*n+9)^10.
a(n) = A011557(A017629(n)). - Wesley Ivan Hurt, Dec 02 2013
a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Nov 25 2021

A160080 Lodumo_4 of Fibonacci numbers.

Original entry on oeis.org

0, 1, 5, 2, 3, 9, 4, 13, 17, 6, 7, 21, 8, 25, 29, 10, 11, 33, 12, 37, 41, 14, 15, 45, 16, 49, 53, 18, 19, 57, 20, 61, 65, 22, 23, 69, 24, 73, 77, 26, 27, 81, 28, 85, 89, 30, 31, 93, 32, 97, 101, 34, 35, 105, 36, 109, 113, 38, 39, 117, 40, 121, 125, 42, 43, 129, 44, 133, 137, 46

Offset: 0

Philippe Deléham, May 01 2009

Permutation of nonnegative numbers.

OEIS wiki, Lodumo transform
Index entries for sequences that are permutations of the nonnegative integers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A004767, A008586, A016825, A017533, A017581, A017629, A079343.

a(n) = lod_4(A000045(n)).
From Philippe Deléham, Nov 30 2023: (Start)
a(n) = 2*a(n-6) - a(n-12) for n >= 12.
a(6*n) = 4*n, a(6*n+1) = 12*n+1, a(6*n+2) = 12*n+5, a(6*n+3) = 4*n+2, a(6*n+4) = 4*n+3, a(6*n+5) = 12*n+9.
G.f.: (x + 5*x^2 + 2*x^3 + 3*x^4 + 9*x^5 + 4*x^6 + 11*x^7 + 7*x^8 + 2*x^9 + x^10 + 3*x^11) / ((1-x)^2*(1+x+x^2)^2*(1+x^3)^2). (End)

A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

Original entry on oeis.org

9, 21, 49, 33, 77, 121, 45, 105, 165, 225, 57, 133, 209, 285, 361, 69, 161, 253, 345, 437, 529, 81, 189, 297, 405, 513, 621, 729, 93, 217, 341, 465, 589, 713, 837, 961, 105, 245, 385, 525, 665, 805, 945, 1085, 1225, 117, 273, 429, 585, 741, 897, 1053, 1209, 1365, 1521

Offset: 0

Vincenzo Librandi, Apr 29 2014

A016838(n) first diagonal.
A085027(n) second diagonal.
A017629(n) column k=0.
Row sums give the second bisection of A002414: 9, 70, 231, 540, 1045, 1794, 2835, 4216, ... [Bruno Berselli, May 08 2014]

			Triangle begins:
n\k |   0     1     2    3    4    5     6     7     8     9
----|--------------------------------------------------------
0   |   9;
1   |  21,   49;
2   |  33,   77,  121;
3   |  45,  105,  165, 225;
4   |  57,  133,  209, 285, 361;
5   |  69,  161,  253, 345, 437, 529;
6   |  81,  189,  297, 405, 513, 621,  729;
7   |  93,  217,  341, 465, 589, 713,  837,  961;
8   | 105,  245,  385, 525, 665, 805,  945, 1085, 1225;
9   | 117,  273,  429, 585, 741, 897, 1053, 1209, 1365, 1521;
.....

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A002145, A004767, A016838, A017629, A085027.

[(4*n+3)*(4*k+3): k in [0..n], n in [0..15]]; /* or, as triangle: */ [[(4*n+3)*(4*k+3): k in [0..n]]: n in [0..10]];

t[n_, k_] := (4 n + 3) (4 k + 3); Table[t[n, k], {n, 0, 10}, {k, n}] // Flatten

Edited by Alois P. Heinz and Bruno Berselli, May 08 2014

A330885 Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, -1, 1, 4, 5, 0, -3, 1, 5, 7, 1, -5, -3, 1, 6, 9, 2, -7, -8, 1, 1, 7, 11, 3, -9, -13, -3, 7, 1, 8, 13, 4, -11, -18, -7, 10, 9, 1, 9, 15, 5, -13, -23, -11, 13, 21, 1, 1, 10, 17, 6, -15, -28, -15, 16, 33, 14, -15

Offset: 0

Bob Selcoe, May 05 2020

			Array starts:
1  1   1  -1   -3   -3    1   7   9   1  -15   -25
1  2   3   0   -5   -8   -3  10  21  14  -17   -52
1  3   5   1   -7  -13   -7  13  33  27  -19   -79
1  4   7   2   -9  -18  -11  16  45  40  -21  -106
1  5   9   3  -11  -23  -15  19  57  53  -23  -133
1  6  11   4  -13  -28  -19  22  69  66  -25  -160
1  7  13   5  -15  -33  -23  25  81  79  -27  -187

Cf. A078016, A180735.

Columns k: A000012 (k=0), A000027 (k=1), A005408 (k=2), A023443 (k=3), A165747 (k=4), -A016885 (k=5), -A004767 (k=6), A016777 (k=7), A017629 (k=8), A190991 (k=9).

T[n_, k_]:= T[n, k]= If[k<3, k*n+1, T[n, k-1] - T[n, k-2] - T[n, k-3]];
Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2020 *)

T(0,k) = A180735(k-1).

T(n,k) - T(n-1,k) = -A078016(k+1).

A360927 Expansion of the g.f. x(1 + 3x + 4x^2 + 4x^3)/((1 - x)^2*(1 + x)).

Original entry on oeis.org

0, 1, 4, 9, 16, 21, 28, 33, 40, 45, 52, 57, 64, 69, 76, 81, 88, 93, 100, 105, 112, 117, 124, 129, 136, 141, 148, 153, 160, 165, 172, 177, 184, 189, 196, 201, 208, 213, 220, 225, 232, 237, 244, 249, 256, 261, 268, 273, 280, 285, 292, 297, 304, 309, 316, 321, 328

Offset: 0

Stefano Spezia, Feb 25 2023

The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the perimeter and the two diagonals of the square.

			Illustrations for n = 1..8:
      o          o o          o o o
                 o o          o o o
                              o o o
  a(1) = 1    a(2) = 4      a(3) = 9
   o o o o    o o o o o    o o o o o o
   o o o o    o o   o o    o o     o o
   o o o o    o   o   o    o   o o   o
   o o o o    o o   o o    o   o o   o
              o o o o o    o o     o o
                           o o o o o o
  a(4) = 16   a(5) = 21     a(6) = 28
   o o o o o o o       o o o o o o o o
   o o       o o       o o         o o
   o   o   o   o       o   o     o   o
   o     o     o       o     o o     o
   o   o   o   o       o     o o     o
   o o       o o       o   o     o   o
   o o o o o o o       o o         o o
                       o o o o o o o o
     a(7) = 33            a(8) = 40

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

Cf. A194274, A016777, A017569, A017629.

LinearRecurrence[{1,1,-1},{0,1,4,9,16},57]

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.
a(0) = 0, a(1) = 1, a(n) = 6*n - 8 for n even, and a(n) = 6*n - 9 for n odd.
E.g.f.: 4*(x + 2) + (6*x - 8)*cosh(x) + (6*x - 9)*sinh(x).
a(2*n) = A017569(n-1) = 4*A016777(n-1).
a(2*n+1) = A017629(n-1).

A382534 Number of minimum total dominating sets in the n-flower graph.

Original entry on oeis.org

3, 9, 9, 36, 15, 81, 21, 36, 27, 225, 33, 36, 39, 441, 45, 36, 51, 729, 57, 36, 63, 1089, 69, 36, 75, 1521, 81, 36, 87, 2025, 93, 36, 99, 2601, 105, 36, 111, 3249, 117, 36

Offset: 1

Eric W. Weisstein, Mar 30 2025

The flower graph is defined for n >= 5. Sequence extended to n = 1 using the formula.

Eric Weisstein's World of Mathematics, Flower Graph.
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A017557 (4-section), A016946 (4-section), A017629 (4-section).

Table[Piecewise[{{36, Mod[n, 4] == 0}, {3 n, Mod[n, 2] == 1}, {9 n^2/4, Mod[n, 4] == 2}}], {n, 20}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {3, 9, 9, 36, 15, 81, 21, 36, 27, 225, 33, 36}, 20]

a(n) = 3*a(n-4)-3*a(n-8)+a(n-12).

G.f.: -3*x*(1+3*x+3*x^2+12*x^3+2*x^4+18*x^5-2*x^6-24*x^7-3*x^8+3*x^9-x^10+12*x^11)/(x-1)^3/(1+x)^3/(x^2+1)^3 . - R. J. Mathar, Apr 02 2025

cp's OEIS Frontend

A096025 Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 6 and (n+7) mod 9 <> 1.

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A096027 Numbers k such that (k+j) mod (2+j) = 1 for j from 0 to 10 and (k+11) mod 13 <> 1.

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A195319 Three times second hexagonal numbers: 3*n*(2*n+1).

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A127989 a(n) = 2*n^3 - 2*n + 9.

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A017638 a(n) = (12n+9)^10.

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A160080 Lodumo_4 of Fibonacci numbers.

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A241747 Triangle read by rows: T(n,k) = (4*n+3)*(4*k+3).

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A195319 Three times second hexagonal numbers: 3n(2*n+1).

A127989 a(n) = 2n^3 - 2n + 9.

A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

A360927 Expansion of the g.f. x(1 + 3x + 4x^2 + 4x^3)/((1 - x)^2*(1 + x)).