A102083 -id:A102083 - cp's OEIS frontend

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A252983 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 1, 1, 3, 6, 13, 1, 13, 6, 10, 41, 33, 33, 41, 10, 15, 85, 266, 68, 266, 85, 15, 21, 145, 851, 1247, 1247, 851, 145, 21, 28, 221, 1836, 8487, 4657, 8487, 1836, 221, 28, 36, 313, 3221, 27905, 67537, 67537, 27905, 3221, 313, 36, 45, 421, 5006, 62977

Offset: 1

R. H. Hardin, Dec 25 2014

Table starts
..0...0....1......3.......6........10..........15............21.............28
..0...0....1.....13......41........85.........145...........221............313
..1...1....1.....33.....266.......851........1836..........3221...........5006
..3..13...33.....68....1247......8487.......27905.........62977.........114433
..6..41..266...1247....4657.....67537......433401.......1481460........3510600
.10..85..851...8487...67537....432842.....5672484......36112108......129234988
.15.145.1836..27905..433401...5672484....60650883.....766674140.....4970634131
.21.221.3221..62977.1481460..36112108...766674140...13458882036...170090480091
.28.313.5006.114433.3510600.129234988..4970634131..170090480091..4857082197177
.36.421.7191.182273.6637020.322183180.18692194423.1139074556531.62656851440792

			Some solutions for n=4 k=4
..0..0..1..1....0..0..0..0....0..0..0..1....0..0..0..0....0..0..1..1
..0..1..1..1....0..0..0..0....0..0..0..1....0..0..0..1....0..1..1..1
..0..1..1..1....0..0..1..1....0..0..0..1....0..0..1..1....1..1..1..1
..0..1..1..1....0..1..1..1....0..0..1..1....0..0..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..480

Column 1 is A000217(n-2)

Column 2 is A102083(n-3)

Empirical for column k:
k=1: a(n) = (1/2)*n^2 - (3/2)*n + 1
k=2: a(n) = 8*n^2 - 44*n + 61 for n>2
k=3: a(n) = 200*n^2 - 1615*n + 3341 for n>4
k=4: a(n) = 8192*n^2 - 87808*n + 241153 for n>6
k=5: a(n) = 557568*n^2 - 7467372*n + 25553940 for n>8
k=6: a(n) = 63438848*n^2 - 1019729920*n + 4176308004 for n>10
k=7: a(n) = 12103190528*n^2 - 226960984822*n + 1081747760523 for n>12

A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 4, 6, 11, 13, 15, 22, 24, 26, 28, 37, 39, 41, 43, 45, 56, 58, 60, 62, 64, 66, 79, 81, 83, 85, 87, 89, 91, 106, 108, 110, 112, 114, 116, 118, 120, 137, 139, 141, 143, 145, 147, 149, 151, 153, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 254, 256, 258, 260, 262, 264, 266, 268, 270, 272, 274, 276, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 352, 354, 356, 358, 360, 362, 364, 366, 368, 370, 372, 374, 376, 378

Offset: 1

Clark Kimberling, Feb 05 2011

This is one of four polka dot arrays in the natural number array A000027:
(odd,odd): A185868
(odd,even): A185869
(even,odd): A185870
(even,even): A185871
row 1: A084849
col 1: A000384
col 2: A091823
diag (1,13,...): A102083
diag (4,24,...): A085250
antidiagonal sums: A059722

			The natural number array A000027 has northwest corner
  1...2...4...7...11
  3...5...8...12..17
  6...9...13..18..24
  10..14..19..25..32
  15..20..26..33..41
The numbers in (odd,odd) positions comprise A185868:
  1....4....11...22...37
  6....13...24...39...58
  15...26...41...60...83
  28...43...62...85...112

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027 (as an array), A185872, A185869, A185870, A185871.

f[n_,k_]:=2n-1+(n+k-2)(2n+2k-3);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import isqrt, comb
def A185868(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-7)+x*(c-5)+5 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2*n-1+(n+k-2)*(2*n+2*k-3).

A360418 Numbers k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class a' mod b' (with r' in {1,...,m'}) iff m < m' or r > r'.

Original entry on oeis.org

1, 2, 3, 5, 13, 17, 20, 25, 41, 48, 53, 61, 85, 95, 102, 113, 145, 158, 167, 181, 221, 237, 248, 265, 313, 332, 345, 365, 421, 443, 458, 481, 545, 570, 587, 613, 685, 713, 732, 761, 841, 872, 893, 925, 1013, 1047, 1070, 1105, 1201, 1238, 1263, 1301, 1405, 1445, 1472, 1513, 1625, 1668, 1697, 1741, 1861

Offset: 1

James Propp, Feb 06 2023

The sequence appears to be the interleaving of the four sequences A080856, A102083, A360416, A360417. This has been verified for values of k up to one million as of February 06 2023.

Above conjecture confirmed with more terms and linear recurrence. See supporting formula below. - Ray Chandler, Feb 10 2025

			The 1st congruence class in the list (with m=1 and r=1) is {1,2,3,...} which contains 1, so 1 is in the sequence. The 2nd congruence class (with m=2 and r=2) is {2,4,6,...} which contains 2, so 2 is in the sequence. The 3rd congruence class (with m=2 and r=1) is {1,3,5,...} which contains 3, so 3 is in the sequence. The 4th congruence class (with m=3 and r=3) is {3,6,9,...} which does not contain 4, so 4 is not in the sequence.

Ray Chandler, Table of n, a(n) for n = 1..14142 (terms up to 10^8)
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2, 0, 0, -1, 1).

Cf. A002024, A004736, A007607, A080856, A102083, A360416, A360417.

mval[n_] := Floor[Sqrt[2 n] + 1/2]; (* A002024 *)
rval[n_] := (2 - 2 n + Round[Sqrt[2 n]] + Round[Sqrt[2 n]]^2)/2; (* A004736 *)
test[n_] := Mod[n - rval[n], mval[n]] == 0;
Select[Range[10000], test[#] &]

From Ray Chandler, Feb 10 2025: (Start)
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.
A080856(n) = A360417(n-1) + 2*A080856(n-1) - 2*A360417(n-2) - A080856(n-2) + A360417(n-3).
A102083(n) = A080856(n) + 2*A102083(n-1) - 2*A080856(n-1) - A102083(n-2) + A080856(n-2).
A360416(n) = A102083(n) + 2*A360416(n-1) - 2*A102083(n-1) - A360416(n-2) + A102083(n-2).
A360417(n) = A360416(n) + 2*A360417(n-1) - 2*A360416(n-1) - A360417(n-2) + A360416(n-2). (End)

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

A337822 Permutation of the natural numbers formed by numbering an infinite square grid by a counterclockwise diamond spiral and visiting them by a counterclockwise stair step spiral.

Original entry on oeis.org

1, 2, 8, 3, 9, 20, 10, 21, 11, 4, 12, 5, 13, 14, 6, 15, 7, 16, 30, 17, 31, 18, 32, 19, 33, 52, 34, 53, 35, 54, 36, 55, 37, 22, 38, 23, 39, 24, 40, 25, 41, 42, 26, 43, 27, 44, 28, 45, 29, 46, 68, 47, 69, 48, 70, 49, 71, 50, 72, 51, 73, 100, 74, 101, 75, 102, 76

Offset: 1

Mohammed Yaseen, Sep 24 2020

nonn

			The path begins:
.
               ---33
                   |
              34  19--32
                       |
          35  20---9  18--31  48
               |   |       |
      36  21--10   3---8  17--30  47
           |           |       |
  37  22  11---4   1---2   7--16  29  46
               |           |
      38  23  12---5   6--15  28  45
                   |   |
          39  24  13--14  27  44
.
              40  25  26  43
.
                  41  42
.

Cf. A102083.
Diamond spiral coordinates (but 0-based): A010751, A305258.
Same type of visit on other types of spirals: A334619, A337838.

a(A102083(n)) = A102083(n) and a(A102083(n)+1) = A102083(n)+1.

A354595 a(n) = n^2 + 4*floor(n/2)^2.

Original entry on oeis.org

0, 1, 8, 13, 32, 41, 72, 85, 128, 145, 200, 221, 288, 313, 392, 421, 512, 545, 648, 685, 800, 841, 968, 1013, 1152, 1201, 1352, 1405, 1568, 1625, 1800, 1861, 2048, 2113, 2312, 2381, 2592, 2665, 2888, 2965, 3200, 3281, 3528, 3613, 3872

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A139098, the second bisection is A102083.

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A102083, A139098, A247375.

Cf. A213037, A327259 (diagonal), A354594, A354596.

a[n_] := n^2 + 4 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354595 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 8, 13, 32}, 60]

PARI
```
a(n) = n^2 + 4*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 4*A008794(n).
G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: 2*x^2*cosh(x) + (1 + 2*x + 2*x^2)*sinh(x). - Stefano Spezia, Jun 07 2022

A103776 Primes p such that 8p^2 + 4p + 1 is also prime.

Original entry on oeis.org

2, 7, 11, 17, 41, 61, 101, 167, 227, 257, 281, 337, 347, 367, 397, 401, 461, 467, 487, 541, 571, 587, 601, 631, 641, 647, 661, 691, 701, 761, 857, 947, 971, 977, 997, 1021, 1087, 1237, 1277, 1291, 1381, 1451, 1481, 1607, 1621, 1627, 1667, 1697, 1787, 1811

Offset: 1

Zak Seidov, Feb 15 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A102083.

[n: n in [0..2000] | IsPrime(n) and IsPrime(8*n^2 + 4*n + 1)]; // Vincenzo Librandi, Feb 03 2014

Select[Prime[Range[300]], PrimeQ[8 #^2 + 4 # + 1]&] (* Harvey P. Dale, Jan 04 2011 *)

A381555 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an even length row of triangles.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 13, 16, 4, 1, 19, 41, 24, 4, 1, 26, 85, 85, 32, 4, 1, 34, 155, 231, 145, 40, 4, 1, 43, 259, 532, 489, 221, 48, 4, 1, 53, 406, 1092, 1365, 891, 313, 56, 4, 1, 64, 606, 2058, 3333, 2926, 1469, 421, 64, 4, 1, 76, 870, 3630, 7359, 8294, 5551, 2255, 545, 72, 4

Offset: 0

Craig Knecht, Feb 27 2025

The total number of ways the diamond can cover a single row of length(n) triangles is the Fibonacci series. This total can be subdivided into categories based on the number of covering diamonds. The number of categories increase with the length of the row providing the structure of the triangle (see illustrations in the link below).
The above process provides a way to subdivide the individual Fibonacci numbers.
Comparing the diamond covering of a row of triangles shown here to the diamond corona of a hexagon A380346 or a diamond A380416 may be instructive.
A381552 provides additional graphics that help explain the diamond covering.

			Triangle begins:
  1, 4;
  1, 8, 4;
  1, 13, 16, 4;
  1, 19, 41, 24, 4;
  1, 26, 85, 85, 32, 4;
  1, 34, 155, 231, 145, 40, 4;

Craig Knecht, Diamond covering of a even length row of triangles.
Craig Knecht, Geometric bisection of the Fibonacci sequence.
Craig Knecht, Subdividing the individual Fibonacci numbers.
Walter Trump, Diamond covering producing the Fibonacci sequence.

Cf. A063496, A081219, A102083, A323847, A380346, A380416, A381552.

cp's OEIS Frontend

A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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A252983 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value increasing by 0 or 1 with every step right, diagonally se or down.

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A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.

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A360418 Numbers k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class a' mod b' (with r' in {1,...,m'}) iff m < m' or r > r'.

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A185438 a(n) = 8*n^2 - 2*n + 1.

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A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

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Maxima

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A337822 Permutation of the natural numbers formed by numbering an infinite square grid by a counterclockwise diamond spiral and visiting them by a counterclockwise stair step spiral.

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A354595 a(n) = n^2 + 4*floor(n/2)^2.

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A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

A185438 a(n) = 8n^2 - 2n + 1.

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

A103776 Primes p such that 8p^2 + 4p + 1 is also prime.