A047209 -id:A047209 - cp's OEIS frontend

A191737 Dispersion of A047212, (numbers >1 and congruent to 0 or 2 or 4 mod 5), by antidiagonals.

1, 2, 3, 4, 5, 6, 7, 9, 10, 8, 12, 15, 17, 14, 11, 20, 25, 29, 24, 19, 13, 34, 42, 49, 40, 32, 22, 16, 57, 70, 82, 67, 54, 37, 27, 18, 95, 117, 137, 112, 90, 62, 45, 30, 21, 159, 195, 229, 187, 150, 104, 75, 50, 35, 23, 265, 325, 382, 312, 250, 174, 125, 84

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....4....7...12
3....5....9...15...25
6....10....17...29...49
8....14...24...40...67
11...19...32...54...90
13...22...37...62...104

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047219, A047212, A191722, A191727, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=4; c2=5; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047212 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191737 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191737  *)

A191738 Dispersion of A047222, (numbers >1 and congruent to 0 or 2 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 12, 10, 9, 8, 20, 17, 15, 11, 13, 33, 28, 25, 18, 14, 22, 55, 47, 42, 30, 23, 16, 37, 92, 78, 70, 50, 38, 27, 19, 62, 153, 130, 117, 83, 63, 45, 32, 21, 103, 255, 217, 195, 138, 105, 75, 53, 35, 24, 172, 425, 362, 325, 230, 175, 125, 88

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....3....5....8
4....7....12...20...33
6....10...17...28...47
9....15...25...42...70
11...18...30...50...83
14...23...38...63...105

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047209, A047222, A191722, A191728, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; c2=5; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047222 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191738 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191738  *)

A191739 Dispersion of A008854, (numbers >1 and congruent to 0 or 1 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 6, 7, 29, 19, 11, 14, 8, 50, 34, 20, 25, 15, 12, 85, 59, 35, 44, 26, 21, 13, 144, 100, 60, 75, 45, 36, 24, 17, 241, 169, 101, 126, 76, 61, 41, 30, 18, 404, 284, 170, 211, 129, 104, 70, 51, 31, 22, 675, 475, 285, 354, 216, 175, 119

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....4....9....16...29
2....5....10...19...34
3....6....11...20...35
7....14...25...44...75
8....15...26...45...76
12...21...36...61...104

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047221, A008854, A191722, A191729, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=4; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A008854 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191739 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191739  *)

A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 11, 10, 8, 7, 20, 18, 15, 13, 9, 35, 31, 26, 23, 16, 12, 60, 53, 45, 40, 28, 21, 14, 101, 90, 76, 68, 48, 36, 25, 17, 170, 151, 128, 115, 81, 61, 43, 30, 19, 285, 253, 215, 193, 136, 103, 73, 51, 33, 22, 476, 423, 360, 323, 228, 173, 123

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....3....6....11...20
2....5....10...18...31
4....8....15...26...45
7....13...23...40...68
9....16...28...48...81
12...21...36...61...103

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047211, A047220, A191722, A191730, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047220 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
  (* A191740 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191740  *)

A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 10, 11, 7, 8, 17, 20, 12, 15, 9, 30, 35, 21, 26, 16, 13, 51, 60, 36, 45, 27, 22, 14, 86, 101, 61, 76, 46, 37, 25, 18, 145, 170, 102, 127, 77, 62, 42, 31, 19, 242, 285, 171, 212, 130, 105, 71, 52, 32, 23, 405, 476, 286, 355, 217, 176, 120

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....5....10...17
3....6....11...20...35
4....7....12...21...36
8....15...26...45...76
9....16...27...46...77
13...22...37...62...105

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047204, A047217, A191722, A191731, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047217 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191741 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191741  *)

A036666 Numbers k such that 5*k + 1 is a square.

Original entry on oeis.org

0, 3, 7, 16, 24, 39, 51, 72, 88, 115, 135, 168, 192, 231, 259, 304, 336, 387, 423, 480, 520, 583, 627, 696, 744, 819, 871, 952, 1008, 1095, 1155, 1248, 1312, 1411, 1479, 1584, 1656, 1767, 1843, 1960, 2040, 2163, 2247, 2376, 2464, 2599, 2691

Offset: 1

N. J. A. Sloane, Dec 11 1999

Third differences are 4, -6, 8, -10, 12, -14, 16, -18, 20, -22, 24, -26, 28, ...
X values of solutions to the equation 5*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007
Also, numbers 5*i^2 + 2*i for integer i. The characteristic function is A205633(n). - Jason Kimberley, Nov 15 2012
From Gary W. Adamson, Sep 22 2019: (Start)
Match the values a(n) with the squares 5k + 1 as follows:
   3,....7,....16,....24,...  .a, a, a, a,...
  16,...36,....81,...121,...  (base).
Then 1/5 in the matching base is equal to .a, a, a,...
Example: 1/5 in base 36 is equal to .7, 7, 7, 7...
Check: 7/36 + 7/36^2 = 259/1296 = .199845...; close to 1/5.
(End)

Jason Kimberley, Table of n, a(n) for n = 1..2000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See D(q).
Ralf Stephan, On the solutions to 'px+1 is square'.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001082, A001622, A002378, A005563, A046092, A047209, A109043, A205633.

List([1..50],n->(10*n*(n-1)+(2*n-1)*(-1)^n+1)/8); # Muniru A Asiru, Nov 28 2018

[(n-1)^2+(n-1)+Ceiling((n-1)/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Jun 05 2014

seq(n^2+n+ceil(n/2)^2, n=0..46); # Gary Detlefs, Feb 23 2010

(Select[ Range[121], Mod[ #, 5] == 1 || Mod[ #, 5] == 4 &]^2 - 1)/5 (* Robert G. Wilson v, Jun 23 2004 *)
Flatten[Position[5*Range[0,3000]+1,?(IntegerQ[Sqrt[#]]&)]]-1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{0,3,7,16,24},50] (* _Harvey P. Dale, Feb 13 2018 *)
Accumulate[Table[n + LCM[n, 2], {n, 0, 121}]] (* Jon Maiga, Nov 28 2018 *)

PARI
```
a(n)=n^2+n+ceil(n/2)^2
```

G.f.: x*(3 + 4*x + 3*x^2) / ((1 - x)*(1 - x^2)).
a(n) has the form ((5*m + 1)^2 - 1)/5 if n is odd; a(n) has the form ((5*m + 4)^2 - 1)/5 if n is even.
a(2*k) = k*(5*k + 2), a(2*k + 1) = 5*k^2 + 8*k + 3. - Mohamed Bouhamida, Nov 06 2007
a(n+1) = n^2 + n + ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010
From Bruno Berselli, Nov 27 2010: (Start)
a(n) = (10*n*(n - 1)+(2*n - 1)*(-1)^n + 1)/8.
5*a(n) + 1 = A047209(n)^2. (End)
a(n) = Sum_{k=0..n} k + A109043(k). - Jon Maiga, Nov 28 2018
E.g.f.: (exp(x)*(1 + 10*x^2) - exp(-x)*(1 + 2*x))/8. - Franck Maminirina Ramaharo, Nov 29 2018
From Amiram Eldar, Mar 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/2.
Sum_{n>=2} (-1)^n/a(n) = 5*(log(5)-1)/4 - sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). (End)

Better description and additional formula from Santi Spadaro, Jul 12 2001

A091998 Numbers that are congruent to {1, 11} mod 12.

Original entry on oeis.org

1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97, 107, 109, 119, 121, 131, 133, 143, 145, 155, 157, 167, 169, 179, 181, 191, 193, 203, 205, 215, 217, 227, 229, 239, 241, 251, 253, 263, 265, 275, 277, 287, 289, 299, 301, 311, 313, 323, 325, 335

Offset: 1

Ray Chandler, Feb 21 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h and n in A000027), then ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 12). Also a(n)^2 - 1 == 0 (mod 24).

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

First row of A092260.
Cf. A175885 (n == 1 or 10 (mod 11)), A175886 (n == 1 or 12 (mod 13)).
Cf. A097933 (primes), A195143 (partial sums).
Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A113801, A175887, A188887.

a091998 n = a091998_list !! (n-1)
a091998_list = 1 : 11 : map (+ 12) a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];

LinearRecurrence[{1,1,-1},{1,11,13},100] (* Harvey P. Dale, Jul 26 2017 *)

is(n)=n=n%12;n==11 || n==1 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 12*n - a(n-1) - 12 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010
a(n) = 6*n + 2*(-1)^n - 3.
G.f.: x*(1+10*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
a(n) = a(n-2) + 12 for n > 2.
a(n) = 12*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 + sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + (6*x - 3)*exp(x) + 2*exp(-x). - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2 + sqrt(3)) = 2*cos(Pi/12) (A188887).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/3)*cos(Pi/12). (End)

Formulae and comment added by Bruno Berselli, Nov 17 2010 - Nov 18 2010

A175885 Numbers that are congruent to {1, 10} mod 11.

Original entry on oeis.org

1, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 100, 109, 111, 120, 122, 131, 133, 142, 144, 153, 155, 164, 166, 175, 177, 186, 188, 197, 199, 208, 210, 219, 221, 230, 232, 241, 243, 252, 254, 263, 265, 274, 276, 285, 287, 296, 298

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 11).

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

Cf. A090771 (n==1 or 9 mod 10), A091998 (n==1 or 11 mod 12).
Cf. A195043 (partial sums).
Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A113801, A175886, A175887, A195312, A195313.

a175885 n = a175885_list !! (n-1)
a175885_list = 1 : 10 : map (+ 11) a175885_list
-- Reinhard Zumkeller, Jan 07 2012

[(22*n+7*(-1)^n-11)/4: n in [1..60]]; // Vincenzo Librandi, Sep 19 2011

Rest[Flatten[{#-1,#+1}&/@(11 Range[0,50])]] (* Harvey P. Dale, Nov 05 2010 *)

a(n)=n%2*9 + 1 \\ Charles R Greathouse IV, Aug 01 2016

G.f.: x*(1+9*x+x^2)/((1+x)*(1-x)^2).
a(n) = (22*n + 7*(-1)^n - 11)/4.
a(n) = -a(-n+1) = a(n-2) + 11 = a(n-1) + a(n-2) - a(n-3).
a(n) = 11*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
a(n) = A195312(n) + A195312(n-1) = A195313(n) - A195313(n-2). - Bruno Berselli, Sep 18 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/11)*cot(Pi/11). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((22*x - 11)*exp(x) + 7*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/11).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/11)*cosec(Pi/11). (End)

A047207 Numbers that are congruent to {0, 1, 3, 4} mod 5.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84

Offset: 1

N. J. A. Sloane

Numbers not ending in 2 or 7. - Bruno Berselli, Oct 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A001622, A030308, A047209, A047218.

[n : n in [0..100] | n mod 5 in [0, 1, 3, 4]]; // Wesley Ivan Hurt, May 30 2016

seq(floor((5*n-3)/4), n=1..57); # Gary Detlefs, Mar 06 2010

Flatten[Table[5*n + {0, 1, 3, 4}, {n, 0, 20}]] (* T. D. Noe, Nov 12 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,4,5},100] (* Harvey P. Dale, Jan 31 2022 *)

forstep(n=0,99,[1,2,1,1],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = floor((5*n-3)/4). - Gary Detlefs, Mar 06 2010
G.f.: x^2*(1 + 2*x + x^2 + x^3) / ( (1 + x)*(x^2 + 1)*(x - 1)^2 ). - R. J. Mathar, Oct 08 2011
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=3, b(k)=5*2^(k-2) for k>1. - Philippe Deléham, Oct 17 2011
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (10*n-9-i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/8, where i=sqrt(-1).
a(2*k) = A047209(k), a(2*k-1) = A047218(k). (End)
E.g.f.: (4 - sin(x) + cos(x) + (5*x - 4)*sinh(x) + 5*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + 3*sqrt(5)*log(phi)/10 + sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021

A203776 Number of partitions of n into distinct parts 5k+1 or 5k+4.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 5, 3, 3, 5, 7, 7, 6, 5, 7, 11, 11, 8, 8, 12, 15, 15, 13, 12, 16, 22, 22, 18, 18, 24, 30, 31, 27, 26, 33, 42, 43, 37, 37, 47, 57, 58, 53, 52, 63, 78, 80, 71, 72, 88, 103, 106, 99, 98, 116, 139, 142

Offset: 0

Reinhard Zumkeller, Jan 05 2012

nonn

Convolution of A281243 and A280454. - Vaclav Kotesovec, Jan 18 2017

			a(10) = #{9+1, 6+4} = 2;
a(20) = #{19+1, 16+4, 14+6, 11+9, 9+6+4+1} = 5.
1 + x + x^4 + x^5 + x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + x^12 + x^13 + 2*x^14 + ...
q + q^61 + q^241 + q^301 + q^361 + q^421 + q^541 + 2*q^601 + 2*q^661 + q^721 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A047209, A003114.

a203776 = p a047209_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

a[ n_] := SeriesCoefficient[ Product[ (1 + x^(5 k - 1)) (1 + x^(5 k - 4)), {k, Ceiling[ n / 5]}], {x, 0, n}] (* Michael Somos, Mar 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5], {x, 0, n}] (* Michael Somos, Mar 23 2013 *)

{a(n) = polcoeff( prod( k=1, ceil(n / 5), (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)), 1 + x * O(x^n)), n)} /* Michael Somos, Mar 23 2013 */

Expansion of f( x, x^4) / f(-x^5, -x^10) in powers of x where f() is the Ramanujan two-variable theta function. - Michael Somos, Mar 23 2013
Expansion of (-x; x^5)oo (-x^4; x^5)_oo in powers of x where (x; q)_oo is the q-Pochhammer symbol. - _Michael Somos, Mar 23 2013
Euler transform of period 10 sequence [ 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, ...]. - Michael Somos, Mar 23 2013
G.f.: Product_{k>0} (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)). - Michael Somos, Mar 23 2013
a(n) ~ exp(sqrt(2*n/15)*Pi) / (2*30^(1/4)*n^(3/4)) * (1 + (Pi/(60*sqrt(30)) - 3*sqrt(15/2)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

cp's OEIS Frontend

A191737 Dispersion of A047212, (numbers >1 and congruent to 0 or 2 or 4 mod 5), by antidiagonals.

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A191738 Dispersion of A047222, (numbers >1 and congruent to 0 or 2 or 3 mod 5), by antidiagonals.

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A191739 Dispersion of A008854, (numbers >1 and congruent to 0 or 1 or 4 mod 5), by antidiagonals.

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A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

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A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

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A036666 Numbers k such that 5*k + 1 is a square.

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A091998 Numbers that are congruent to {1, 11} mod 12.

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