A033991 -id:A033991 - cp's OEIS frontend

A033990 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative y-axis.

0, 1, 1, 8, 3, 7, 6, 2, 1, 5, 1, 1, 6, 2, 2, 1, 3, 4, 0, 4, 5, 3, 6, 7, 0, 8, 9, 1, 4, 6, 1, 2, 7, 1, 1, 4, 4, 8, 1, 7, 4, 7, 2, 0, 8, 8, 2, 4, 4, 1, 2, 8, 4, 6, 3, 2, 7, 3, 3, 7, 3, 2, 4, 1, 2, 3, 4, 7, 5, 6, 5, 2, 0, 1, 5, 8, 9, 8, 6, 4, 1, 7, 6, 1, 7, 8, 7, 7, 5, 1, 8, 4, 7, 6, 9, 2, 2, 3, 9, 0, 1, 0, 1, 6, 8

Offset: 0

N. J. A. Sloane

Consider array of digits 0_(1)23456789(1)0111213141516171(8)1920212223...; in this array add to n-th pointer 8*n+1 to get next pointer. E.g., n=1 so n+(8*1+1)=10 -> n=10 so n+(8*2+1)=27 -> n=27 so ... etc. - comment from Patrick De Geest.

			The spiral begins
                 2---3---2---4---2---5---2
                 |                       |
                 2   1---3---1---4---1   6
                 |   |               |   |
                 2   2   4---5---6   5   2
                 |   |   |       |   |   |
                 1   1   3   0   7   1   7
                 |   |   |   |   |   |   |
                 2   1   2---1   8   6   2
                 |   |           |   |   |
                 0   1---0---1---9   1   8
                 |                   |   |
                 2---9---1---8---1---7   2
                                         |
                             3---0---3---9
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading downwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033989. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 - 3n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2-3*n-1) for n > 0. - Andrew Woods, May 20 2012

More terms from Patrick De Geest, Oct 15 1999

Edited by Charles R Greathouse IV, Nov 01 2009

A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

Original entry on oeis.org

0, 7, 1, 7, 4, 2, 8, 1, 1, 3, 1, 2, 0, 2, 3, 1, 3, 4, 6, 5, 5, 5, 7, 7, 8, 8, 9, 6, 8, 1, 1, 1, 2, 3, 1, 8, 0, 6, 1, 7, 0, 9, 2, 8, 4, 3, 2, 1, 1, 7, 2, 6, 2, 1, 3, 3, 5, 5, 3, 2, 2, 0, 4, 3, 2, 5, 4, 6, 5, 0, 5, 1, 1, 6, 5, 8, 1, 2, 6, 7, 3, 8, 7, 8, 9, 5, 7, 1, 8, 2, 8, 6, 1, 9, 9, 3, 6, 7, 9, 0, 1, 4, 6, 1, 0

Offset: 0

N. J. A. Sloane

			  2---3---2---4---2---5---2
  |                       |
  2   1---3---1---4---1   6
  |   |               |   |
  2   2   4---5---6   5   2
  |   |   |       |   |   |
  1   1   3   0   7   1   7
  |   |   |   |   |   |   |
  2   1   2---1   8   6   2
  |   |           |   |   |
  0   1---0---1---9   1   8
  |                   |   |
  2---9---1---8---1---7   2
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading rightwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Sequences based on the same spiral: A033988, A033989, A033990. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]];
a[n_] := If[n==0, 0, A033307[[4n^2 + 3n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + 3*n - 1) for n > 0. - Andrew Woods, May 20 2012

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A028994 Even 10-gonal (or decagonal) numbers.

Original entry on oeis.org

0, 10, 52, 126, 232, 370, 540, 742, 976, 1242, 1540, 1870, 2232, 2626, 3052, 3510, 4000, 4522, 5076, 5662, 6280, 6930, 7612, 8326, 9072, 9850, 10660, 11502, 12376, 13282, 14220, 15190, 16192, 17226, 18292, 19390, 20520, 21682, 22876, 24102, 25360, 26650, 27972

Offset: 0

Patrick De Geest

a(n) (for n >= 1) is also the Wiener index of the windmill graph D(5, n). The windmill graph D(m, n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m, n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3, n), D(4, n), and D(6, n) see A033991, A152743, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Decagonal Number.
Eric Weisstein's World of Mathematics, Windmill Graph. - _Emeric Deutsch_, Sep 21 2010
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A028993, A033991, A139273, A152743, A180577.

[2*n*(8*n - 3): n in [0..60]]; // Vincenzo Librandi, Oct 18 2013

CoefficientList[Series[-2 x (11 x + 5)/(x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{3, -3, 1}, {0, 10, 52}, 40] (* Harvey P. Dale, Dec 10 2014 *)
Table[16n^2 - 6n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)

a(n)=2*n*(8*n-3) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n*(8*n - 3). - Omar E. Pol, Aug 19 2011
G.f.: -2*x*(11*x+5)/(x-1)^3. - Colin Barker, Nov 18 2012
Sum_{n>=1} 1/a(n) = (8*log(2) - (sqrt(2)-1)*Pi - 2*sqrt(2)*log(1+sqrt(2)))/12. - Amiram Eldar, Feb 27 2022
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: 2*x*(5 + 8*x)*exp(x).
a(n) = 2*A139273(n) = A001107(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A113688 Isolated semiprimes in the semiprime square spiral.

Original entry on oeis.org

65, 74, 249, 295, 309, 355, 422, 511, 545, 667, 669, 758, 926, 943, 979, 998, 1099, 1167, 1186, 1322, 1457, 1469, 1561, 1585, 1658, 1711, 1774, 1779, 1835, 1891, 1959, 1961, 1963, 2021, 2038, 2066, 2155, 2186, 2191, 2206, 2271, 2329, 2342

Offset: 1

Jonathan Vos Post, Nov 05 2005

Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam's marking the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by marking the semiprimes (A001358). Each integer has 8 adjacent integers in the spiral, horizontally, vertically and diagonally. Curious extended clumps coagulate, slightly denser towards the origin, of semiprimes connected by adjacency. This sequence lists the isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes. A113689 gives an enumeration of the number of semiprimes in clumps of size > 1 through n^2.

The squares of twin primes occupy adjacent points along the southeast diagonal, so none are isolated. Thus the only isolated semiprimes in the spiral that are squares are the squares of "isolated primes" (A007510). The first square in this sequence is a(1473) = 66049 = 257^2. - Jon E. Schoenfield, Aug 12 2018

			Spiral example:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 121 showing only semiprimes; the isolated semiprimes appear in parentheses:
.
    .---.---.---.---.---.--95--94--93---.--91
    |                                       |
    . (65)--.---.--62---.---.---.--58--57   .
    |   |                               |   |
    .   .   .---.--35--34--33---.---.   .   .
    |   |   |                       |   |   |
    .   .  38   .---.--15--14---.   .  55   .
    |   |   |   |               |   |   |   |
    .   .  39   .   .---4---.   .   .   .  87
    |   |   |   |   |       |   |   |   |   |
  106  69   .   .   6   .---.   .   .   .  86
    |   |   |   |   |           |   |   |   |
    .   .   .   .   .---.---9--10   .   .  85
    |   |   |   |                   |   |   |
    .   .   .  21--22---.---.--25--26  51   .
    |   |   |                           |   |
    .   .   .---.---.--46---.---.--49---.   .
    |   |                                   |
    .   .-(74)--.---.--77---.---.---.---.--82
    |
  111---.---.---.-115---.---.-118-119---.-121
.
(End)

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.

Michael De Vlieger, Table of n, a(n) for n = 1..2000
Alois P. Heinz, Plot of semiprime spiral, containing all semiprimes <= 10000. Isolated semiprimes are colored red.
M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 43-44, 1967.
M. Stein and S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, Amer. Math. Monthly 71, 516-520, 1964.
Eric Weisstein's World of Mathematics, Prime Spiral.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826.

Cf. A115258 (isolated primes in Ulam's lattice).

spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; f[w_] := Block[{d = Dimensions@ w, t, g}, t = Reap[Do[Sow@ Take[#[[k]], {2, First@ d - 1}], {k, 2, Last@ d - 1}]][[-1, 1]] &@ w; g[n_] := If[n != 0, Total@ Join[Take[w[[Last@ # - 1]], {First@ # - 1, First@ # + 1}], {First@ #, Last@ #} &@ Take[w[[Last@ #]], {First@ # - 1, First@ # + 1}], Take[w[[Last@ # + 1]], {First@ # - 1, First@# + 1}]] &@(Reverse@ First@ Position[t, n] + {1, 1}) == 0, False]; Select[Union@ Flatten@ t, g@ # &]]; t = spiral@ 26 /. n_ /; PrimeOmega@ n != 2 -> 0; f@ t (* Michael De Vlieger, Dec 21 2015, Version 10 *)

Corrected and extended by Alois P. Heinz, Jan 02 2011

A014848 a(n) = n^2 - floor( n/2 ).

Original entry on oeis.org

0, 1, 3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390, 431, 473, 518, 564, 613, 663, 716, 770, 827, 885, 946, 1008, 1073, 1139, 1208, 1278, 1351, 1425, 1502, 1580, 1661, 1743, 1828, 1914, 2003, 2093, 2186, 2280, 2377, 2475

Offset: 0

N. J. A. Sloane

Quasipolynomial of order 2. - Charles R Greathouse IV, Jan 19 2012

The binomial transform is 0, 1, 5, 20,... which is A084850 with offset 1. - R. J. Mathar, Nov 26 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A033951, A033991, A042963 (first differences), A084850.

[n^2-Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Nov 14 2014

A014848:=n->n^2 - floor(n/2); seq(A014848(n), n=0..50); # Wesley Ivan Hurt, Oct 11 2013

Table[n^2-Floor[n/2],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
LinearRecurrence[{2,0,-2,1},{0,1,3,8},60] (* Harvey P. Dale, Jun 13 2022 *)

PARI
```
a(n)=n^2-n\2
```

def A014848(n): return n**2-(n>>1) # Chai Wah Wu, Jan 18 2023

[n^2 - (n//2) for n in range(71)] # G. C. Greubel, Mar 14 2024

a(2*n) = A033991(n).
a(2*n+1) = A033951(n).
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1-x^2)).
a(n) = (2*n*(2*n-1) + 1 - (-1)^n)/4. - Bruno Berselli, Feb 17 2011
a(n) = round(n/(exp(1/n) - 1)), n > 0. - Richard R. Forberg, Nov 14 2014
E.g.f.: (1/4)*((1 + 2*x + 4*x^2)*exp(x) - exp(-x)). - G. C. Greubel, Mar 14 2024

A113689 Number of semiprimes in clumps of size > 1 through n^2 in the semiprime spiral.

Original entry on oeis.org

0, 0, 2, 6, 9, 13, 17, 21, 23, 31, 37, 45, 54, 59, 72, 77, 83, 93, 104, 116, 125, 140, 150, 164, 180, 188, 203, 219, 236, 255, 272, 287, 301, 317, 334, 354, 378, 403, 419, 430, 450, 475, 498, 521, 542, 560, 588, 608, 626, 652, 677, 698

Offset: 1

Jonathan Vos Post, Nov 05 2005

Write the integers 1, 2, 3, 4, ... in a counterclockwise square spiral. Analogous to Ulam coloring in the primes in the spiral and discovering unexpectedly many connected diagonals, we construct a semiprime spiral by coloring in all semiprimes (A001358). Each integer has 8 adjacent integers in the spiral, horizontally, vertically and diagonally. Curious extended clumps coagulate, slightly denser towards the origin, of semiprimes connected by adjacency. This sequence, A113689, gives an enumeration of the number of semiprimes in clumps of size > 1 through n^2, not looking past the square boundary. A113688 gives isolated semiprimes in the semiprime spiral, namely those semiprimes none of whose adjacent integers in the spiral are semiprimes.

			a(3) = 2 because there is one visible clump through 3^2 = 9, {4,6}, which two semiprimes are diagonally connected.
a(4) = 6 because there are 6 semiprimes in the 2 visible clumps through 4^2 = 16, {4, 6, 14, 15}, {9, 10}.
a(5) = 9 because there are 9 semiprimes in the 3 visible clumps through 5^2 = 25, {4, 6, 14, 15}, {9, 10, 25}, {21, 22}.
......................
... 17 16 15 14 13 ...
... 18  5  4  3 12 ...
... 19  6  1  2 11 ...
... 20  7  8  9 10 ...
... 21 22 23 24 25 ...
......................

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.

M. Stein and S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74, 43-44, 1967.
M. Stein, S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, Amer. Math. Monthly 71, 516-520, 1964.
Eric Weisstein's World of Mathematics, Prime Spiral.
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001107, A001358, A002939, A002943, A004526, A005620, A007742, A033951-A033954, A033988, A033989-A033991, A033996, A063826, A113688.

Corrected and extended by Alois P. Heinz, Jan 02 2011

A180577 The Wiener index of the windmill graph D(6,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs).

Original entry on oeis.org

15, 80, 195, 360, 575, 840, 1155, 1520, 1935, 2400, 2915, 3480, 4095, 4760, 5475, 6240, 7055, 7920, 8835, 9800, 10815, 11880, 12995, 14160, 15375, 16640, 17955, 19320, 20735, 22200, 23715, 25280, 26895, 28560, 30275, 32040, 33855, 35720, 37635, 39600, 41615, 43680, 45795

Offset: 1

Emeric Deutsch, Sep 21 2010

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
The Wiener polynomial of D(m,n) is (1/2)n(m-1)t[(m-1)(n-1)t+m].
The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1].
For the Wiener indices of D(3,n), D(4,n), and D(5,n) see A033991, A152743, and A028994, respectively.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
Eric Weisstein's World of Mathematics, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028994, A033991, A152743.

Maple
```
seq(5*n*(-2+5*n), n = 1 .. 40);
```

a(n)=5*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 5*n*(5*n-2).
G.f.: -5*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 30 2012
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 5*exp(x)*x*(3 + 5*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A193866 Even pentagonal numbers divided by 2.

Original entry on oeis.org

0, 6, 11, 35, 46, 88, 105, 165, 188, 266, 295, 391, 426, 540, 581, 713, 760, 910, 963, 1131, 1190, 1376, 1441, 1645, 1716, 1938, 2015, 2255, 2338, 2596, 2685, 2961, 3056, 3350, 3451, 3763, 3870, 4200, 4313, 4661, 4780, 5146, 5271, 5655, 5786, 6188

Offset: 0

Omar E. Pol, Aug 18 2011

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A001105, A033991.

[1/16*(1-3*(-1)^n+12*n)*(1-(-1)^n+4*n): n in [0..60]]; // Vincenzo Librandi, Jun 20 2015

Table[(1/16 (1 - 3 (-1)^n + 12 n) (1 - (-1)^n + 4 n)), {n, 0, 50}] (* Vincenzo Librandi, Jun 20 2015 *)
Select[PolygonalNumber[5,Range[0,100]],EvenQ]/2 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 13 2018 *)

a(n)=3*n^2+if(n%2,5*n+1,-n)/2 \\ Charles R Greathouse IV, Aug 18 2011

a(n) = 1/16*(1-3*(-1)^n+12*n)*(1-(-1)^n+4*n).
a(n) = A014633(n)/2.
a(0)=0, a(1)=6, a(2)=11, a(3)=35, a(4)=46, a(n)=a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). G.f.: x*(6+5*x+12*x^2+x^3)/(1-x-2*x^2+2*x^3+x^4-x^5). [Colin Barker, Jan 25 2012]

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0

Offset: 0

Seiichi Manyama, Mar 01 2021

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  0, 2,   4,    6,     8,    10, ...
  0, 3,  14,   33,    60,    95, ...
  0, 4,  48,  180,   448,   900, ...
  0, 5, 164,  981,  3344,  8525, ...
  0, 6, 560, 5346, 24960, 80750, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
Rows 0..2 give A000012, A005843, A033991.
Main diagonal gives (-1) * A109520(n+1).
Cf. A342120, A342129, A342134.

T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021

T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)

T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));

T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.

A185950 a(n) = 4*n^2 - n - 1.

Original entry on oeis.org

-1, 2, 13, 32, 59, 94, 137, 188, 247, 314, 389, 472, 563, 662, 769, 884, 1007, 1138, 1277, 1424, 1579, 1742, 1913, 2092, 2279, 2474, 2677, 2888, 3107, 3334, 3569, 3812, 4063, 4322, 4589, 4864, 5147, 5438, 5737, 6044, 6359, 6682, 7013, 7352, 7699, 8054, 8417, 8788, 9167, 9554, 9949, 10352, 10763, 11182, 11609

Offset: 0

Paul Curtz, Feb 07 2011

Write the sequence A023443 in a clockwise spiral. a(n) is on the y-axis.

a(n) mod 9 = period 9: repeat [8,2,4,5,5,4,2,8,4] = A182868(n+2) mod 9.

			  11--12--13--14--15
   |               |
  10   1---2---3  16
   |   |       |   |
   9   0-(-1)  4  17
   |           |   |
   8---7---6---5  18

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033951, A182868.

a185950 n = (4 * n - 1) * n - 1 -- Reinhard Zumkeller, Aug 14 2013

[-1-n+4*n^2: n in [0..80]]; // Vincenzo Librandi, Feb 08 2011

A185950:=n->4*n^2-n-1: seq(A185950(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2017

Table[4n^2-n-1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{-1,2,13},60] (* Harvey P. Dale, May 22 2015 *)

a(n)=4*n^2-n-1 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = A176126(4*n-1) = A054556(n+1) - 2 = A033991(n) - 1.
a(n) = a(n-1) + 8*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: ( 1-5*x-4*x^2 ) / (x-1)^3. - R. J. Mathar, Feb 10 2011
E.g.f.: (4*x^2 + 3*x - 1)*exp(x). - G. C. Greubel, Jul 23 2017

cp's OEIS Frontend

A033990 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative y-axis.

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A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

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A028994 Even 10-gonal (or decagonal) numbers.

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A113688 Isolated semiprimes in the semiprime square spiral.

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A014848 a(n) = n^2 - floor( n/2 ).

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A113689 Number of semiprimes in clumps of size > 1 through n^2 in the semiprime spiral.

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A180577 The Wiener index of the windmill graph D(6,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs).

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A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).