A002943 -id:A002943 - cp's OEIS frontend

A125202 a(n) = 4n^2 - 6n + 1.

-1, 5, 19, 41, 71, 109, 155, 209, 271, 341, 419, 505, 599, 701, 811, 929, 1055, 1189, 1331, 1481, 1639, 1805, 1979, 2161, 2351, 2549, 2755, 2969, 3191, 3421, 3659, 3905, 4159, 4421, 4691, 4969, 5255, 5549, 5851, 6161, 6479, 6805, 7139, 7481, 7831, 8189, 8555, 8929, 9311, 9701, 10099, 10505, 10919, 11341

Offset: 1

Reinhard Zumkeller, Nov 24 2006

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

Cf. A125199.
Cf. A003415, A017089.
Cf. A002939, A016789, A017041, A017485, A028387.
Cf. A002943, A028387.

[4*n^2 - 6*n + 1: n in [1..60]]; // Vincenzo Librandi, Jul 11 2011

f[a_]:=4*a^2-6*a+1; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 14 2009 *)

a(n)=4*n^2-6*n+1 \\ Charles R Greathouse IV, Sep 28 2015

a(n) = A125199(n,n-1) for n>1.
A003415(a(n)) = A017089(n-1).
From Arkadiusz Wesolowski, Dec 25 2011: (Start)
a(1) = -1, a(n) = a(n-1) + 8*n - 10.
a(n) = 2*a(n-1) - a(n-2) + 8 with a(1) = -1 and a(2) = 5.
G.f.: (1 - 4*x + 11*x^2)/(1 - x)^3. (End)
a(n) = A002943(n-1) - 1. - Arkadiusz Wesolowski, Feb 15 2012
a(n) = A028387(2n-3), with A028387(-1) = -1. - Vincenzo Librandi, Oct 10 2013
E.g.f.: exp(x)*(1 - 2*x + 4*x^2). - Stefano Spezia, Oct 10 2022
Sum_{n>=1} 1/a(n) = sqrt(5)/10*(psi(1/4+sqrt(5)/4) - psi(1/4-sqrt(5)/4)) = -0.656213833... - R. J. Mathar, Apr 22 2024

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006

The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].
The first difference between the entries in row r is r.
Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013
Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013
a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024

			The array starts, with row r=0, as
  r=0:   0  0  0  0  0  0  0  0;
  r=1:   1  2  3  4  5  6  7  8;
  r=2:  10 12 14 16 18 20 22 24;
  r=3:  27 30 33 36 39 42 45 48;

G. C. Greubel, Table of n, a(n) for n = 0..1000
LeetCode, 3014. Minimum Number of Pushes to Type Word I.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.

Flatten[Table[4r^2+r(Range[-3,4]),{r,0,6}]] (* or *) LinearRecurrence[ {2,-1,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,1,2},60] (* Harvey P. Dale, Nov 26 2015 *)

From Philippe Deléham, Mar 26 2013: (Start)
a(8k)   = A001107(k).
a(8k+1) = A002939(k).
a(8k+2) = A033991(k).
a(8k+3) = A016742(k).
a(8k+4) = A007742(k).
a(8k+5) = A002943(k).
a(8k+6) = A033954(k).
a(8k+7) = A033996(k). (End)
G.f.: x^8/((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013
a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019
a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024

Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010

A003148 a(n+1) = a(n) + 2n(2n+1)a(n-1), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 7, 27, 321, 2265, 37575, 390915, 8281665, 114610545, 2946939975, 51083368875, 1542234996225, 32192256321225, 1114841223671175, 27254953356505875, 1064057291370698625, 29845288035840902625, 1296073464766972266375, 41049997128507054562875

Offset: 0

N. J. A. Sloane

Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are successive powers of 2.
a(n) is the coefficient of x^n in arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) multiplied by (2*n+1)!!.
This sequence is the linking pin between the a(n) formulas of the ED1, ED2, ED3 and ED4 array rows, see A167552, A167565, A167580 and A167591. - Johannes W. Meijer, Nov 23 2009

			arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.
R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, arXiv:math/0306184 [math.NA], 2003-2004; cf. Eq. 22.

Cf. A002943, A046161, A049606, A077568, A084543, A091520, A123746.

Cf. A167552, A167565, A167580, A167591.

a003148 n = a003148_list !! n
a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)
                          (zipWith (*) (tail a002943_list) a003148_list)
-- Reinhard Zumkeller, Nov 22 2011

[n le 2 select 1 else Self(n-1) + 2*(n-2)*(2*n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 04 2022

# double factorial of odd "l" df := proc(l) local n; n := iquo(l,2); RETURN( factorial(l)/2^n/factorial(n)); end: x := 1; for n from 1 to 15 do if n mod 2 = 0 then x := 2*n*x+df(2*n-1); else x := 2*n*x-df(2*n-1); fi; print(x); od; quit

a[n_] := a[n] = (-1)^n*(2n - 1)!! + 2n*a[n - 1]; a[0] = 1; Table[ a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 01 2011, after R. J. Mathar *)
a[ n_] := If[ n < 0, 0, (2 n + 1)!! Hypergeometric2F1[ -n, 1/2, 3/2, 2]]; (* Michael Somos, Apr 20 2018 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 2 x) Sqrt[1 + 2 x]), {x, 0, n}]]; (* Michael Somos, Apr 20 2018 *)
RecurrenceTable[{a[0]==a[1]==1,a[n+1]==a[n]+2n(2n+1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Jul 27 2019 *)

Vec(serlaplace(1/(sqrt(1+2*x + O(x^20))*(1-2*x)))) \\ Andrew Howroyd, Feb 05 2018

@CachedFunction
def a(n): return 1 if (n<2) else a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) # a = A003148
[a(n) for n in range(31)] # G. C. Greubel, Nov 04 2022

a(n) = (-1)^n*(2n-1)!! + 2*n*a(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - R. J. Mathar, Jun 12 2003
a(n) = ((2*n+1)!!/4) * Integral_{-Pi..Pi} cos(x)^n * cos(x/2) dx. - R. J. Mathar, Jun 30 2003
a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - R. J. Mathar, Jun 30 2003
In terms of the (terminating) Gauss hypergeometric function/series, 2F1(., .; .; 2), a(n) is a special case of the family of integer sequences defined by a(m, n) = ((2*n+2*m+1)!!/(2*m+1)) * 2F1(-n, m+1/2; m+3/2; 2), for m >= 0, n >= 0. An integral form can be seen as a(m, n) = ((2*n+2*m+1)!!/4) * Integral_{-Pi..Pi} (sin(x/2))^(2*m) * (cos(x))^n * cos(x/2) dx. A recurrence property is 4*(n+1)*a(m, n) = (2*m-1)*a(m-1, n+1) + (-1)^n*(2*n+2*m+1)!!. Sequences that have these properties are a(0, n) = this sequence, a(1, n) = A077568, a(2, n) = A084543. - R. J. Mathar, Jun 30 2003
E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - Vladeta Jovovic, Oct 12 2003
a(n) = (2^n)*n!*A123746(n)/A046161(n) = (2^n)*n!*Sum_{k=0..n} binomial(2*k,k)*(-1/4)^k. From the e.g.f. - Wolfdieter Lang, Oct 06 2008
a(n) = A049606(n)*A123746(n). - Johannes W. Meijer, Nov 23 2009
a(n) = A091520(n) * n! / 2^n. - Michael Somos, Mar 17 2011

a(16)-a(20) from Andrew Howroyd, Feb 05 2018

A007622 Consider Leibniz's harmonic triangle (A003506) and look at the non-boundary terms. Sequence gives numbers appearing in denominators, sorted.

Original entry on oeis.org

6, 12, 20, 30, 42, 56, 60, 72, 90, 105, 110, 132, 140, 156, 168, 182, 210, 240, 252, 272, 280, 306, 342, 360, 380, 420, 462, 495, 504, 506, 552, 600, 630, 650, 660, 702, 756, 812, 840, 858, 870, 930, 992, 1056, 1092, 1122, 1190, 1260, 1320, 1332

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

nonn

No term is prime, about 80% are abundant, but the first few deficient are: 105, 110, 182, 495, 506, 1365, 1406, 1892, 2162, 2756, 2907, 3422, 3782, 4556, 5313, .... - Robert G. Wilson v, Aug 16 2010

A002943 = (6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, ...) is a subsequence: indeed, this is every second denominator of the first differences of the sequence 1/n. - M. F. Hasler, Oct 11 2015

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.

Robert G. Wilson v, Table of n, a(n) for n = 1..1217.
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle.

L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Union[ Flatten[ Table[ 1/L[n, m], {n, 3, 150}, {m, 2, Floor[n/2 + .5]}]]], 65]
t[n_, k_] := Denominator[n!*k!/(n + k + 1)!]; Take[ DeleteDuplicates@ Rest@ Sort@ Flatten@ Table[t[n - k, k], {n, 2, 150}, {k, n/2 + 1}], 65] (* Robert G. Wilson v, Jun 12 2014 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jul 25 2000. Rechecked Jun 27 2003.

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.

Original entry on oeis.org

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

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

Original entry on oeis.org

1, 3, -1, 15, -10, 1, 105, -105, 21, -1, 945, -1260, 378, -36, 1, 10395, -17325, 6930, -990, 55, -1, 135135, -270270, 135135, -25740, 2145, -78, 1, 2027025, -4729725, 2837835, -675675, 75075, -4095, 105, -1, 34459425, -91891800, 64324260, -18378360, 2552550, -185640, 7140, -136

Offset: 0

Wolfdieter Lang, Jul 13 2007

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.
The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
(End)
Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

			[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
T. Copeland, Juggling Zeros in the Matrix: Example II, 2020.
Wolfdieter Lang, First ten rows and more.

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).
Row sums (signed) give A131441. Row sums (unsigned) give A066224.
Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.

seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2,n-m)/m!,m=0..n),n=0..10); # Robert Israel, Dec 25 2015

Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)

a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015
From Tom Copeland, Dec 14 2015: (Start)
Production matrix is
   3,   -1;
  -6,    7,   -1;
   0,  -20,   11,   -1;
   0,    0,  -42,   15,   -1;
   0,    0,    0,  -72,   19,   -1;
   0,    0,    0,    0, -110,   23,   -1;
   0,    0,    0,    0,    0, -156,   27,   -1;
   0,    0,    0,    0,    0,    0, -210,   31,   -1;
   0,    0,    0,    0,    0,    0,    0, -272,   35,   -1;
  ... (End)

Title formula corrected by Tom Copeland, Dec 12 2015

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

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

A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

3, 11, 23, 83, 131, 179, 191, 239, 243, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 891, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1539, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2211, 2339, 2351, 2399, 2459, 2511, 2543, 2699, 2819, 2903

Offset: 1

Reikku Kulon, Mar 11 2009

Superset of A002515; 2n + 1 is prime. A recursive search for members of this sequence results in the infinite series of very large primes A145918. Most members of this sequence are also prime, but five members less than 10000 are composite:
.. . 243 = 3^5
.. . 891 = 3^4 * 11
. . 1539 = 3^4 * 19
. . 2211 = 3 * 11 * 67
. . 2511 = 3^4 * 31
The polygonal number with f sides of length 2n + 1 is (2^n - 1)(2^(n - 1)).
Contribution from Reikku Kulon, May 19 2009: (Start)
The average difference between successive composite terms gradually increases, remaining near their order of magnitude. Roughly 3% of all primes less than 20 billion belong to this sequence or the 2n + 1 sequence. The interval between composite terms 12228632879 and 13169544651 contains 1113606 primes, accounting for 2.75% of the primes in the interval and 1.42% of the primes between 24457265759 and 26339089303.
Prime factors are most often congruent to 3 (mod 4), but some factors are congruent to 1 (mod 4), especially when a term has an even number of not necessarily distinct factors. The most common factor is 3, and often a large power of 3 is a divisor. 5, 7, 13, and 17 are never factors.
The ones digit of composite terms is most often 1, and becomes progressively more likely to be 1. It is never 5. It cannot be 7, because 2n + 1 would then be divisible by 5. The lack of solutions with n divisible by 5 appears crucial to the consistent primality of 2n + 1.
The tens digit is odd if the ones digit is 1 or 9; it is even if the ones digit is 3. This is a consequence of congruence to 3 (mod 4).
The most common least significant two digits of composite terms are 51.
The least significant digits of prime terms do not follow an obvious distribution.
This is the simplest and possibly most productive member of a family of similar sequences defined by f = (s + 8n^2 - 2) / (2n * (2n + 1)), where s is pronic. For these sequences, 2n + 1 is dominated by primes.
=====================================
Large sequences of consecutive primes
=====================================
.   Initial term    Primes   Predecessor     Successor          Gap
.   ---------------------------------------------------------------
.     1529648303    157285    1529648231    1639846391    110198160
.     3832649339    473045    3832647111    4193496803    360849692
.     5897103683    411434    5897102751    6223464171    326361420
.     6543227423    445293    6543226251    6899473631    356247380
.     8126586971    913506    8126586711    8871331491    744744780
.     9533381219    689395    9533380131   10103115231    569735100
.    11576086883    708712   11576086731   12171829419    595742688
.    12228633251   1113606   12228632879   13169544651    940911772
.    21315457451   2328623   21315457251   23375077119   2059619868
(End)

			ngon(f, k) = k * (f * (k - 1) / 2 - k + 2)
. . . 3 = (4^3 - 2^3 + 8 * 9 - 2) / (6 * 7)
. . . . = (2 * 28 + 70) / 42
. . 126 = (2 * 28 + 70)
.. . 28 = (2^3 - 1) * 2^2
. . . . = 126 - 70 - 28
. . . . = 7 * (18 - 10 - 4)
. . . . = 7 * (3 * 6 - 3 * 3 - 5)
. . . . = 7 * (3 * 3 - 7 + 2)
.. 8287 = (4^11 - 2^11 + 8 * 121 - 2) / (22 * 23)
. . . . = (2 * 2096128 + 966) / 506
4193222 = (2 * 2096128 + 966)
2096128 = (2^11 - 1) * 2^10
. . . . = 4193222 - 2096128 - 966
. . . . = 23 * (182314 - 91136 - 42)
. . . . = 23 * (8287 * 22 - 8287 * 11 - 21)
. . . . = 23 * (8287 * 11 - 23 + 2)
Coincidentally, 8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime, and may be the largest value of f that is.
1031 = 257 * 4 + 3 and 2063 = 1031 * 2 + 1 are both members of this sequence, 4127 = 2063 * 2 + 1 is prime, and 8287 = (4127 + 16) * 2 + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A139601 (polygonal numbers)
Cf. A046318, A139876 (related to composite members 243, 891, 1539, and 2511)
Cf. A060210, A002034, A109833, A136801 (their factors)
Cf. A039506 (3, 8287)
Cf. A006516 (2^n - 1)(2^(n - 1))
Cf. A000051 (Fermat numbers), A019434 (Fermat primes)
Cf. A142291 (prime sequence 257, 1031, 2063, 4127)
Cf. A235540 (nonprimes), A002943.

a158034 n = a158034_list !! (n-1)
a158034_list = [x | x <- [1..],
                    (4^x - 2^x + 8*x^2 - 2) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

Original entry on oeis.org

-1, 0, -3, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837, 870, 899, 930, 957, 992, 1023, 1056, 1085, 1122, 1155, 1190

Offset: 0

Paul Curtz, Jul 11 2012

Let a(n)/A000290(n) = [-1/0, 0/1, -3/4, 2/9, 3/16, 6/25, 5/36, 12/49, 15/64, 20/81, 21/100, 30/121, ...] = a(n)/b(n) (say).
Then b(n)-4*a(n)=4, 1, 16, 1 (period of length 4).
Permutation from a(n) to A061037(n): 1, 3, 2, 7, 5, 11, 4, 15, 9, 19, 6, ... = shifted A145979 + 1.
A061037(n) - a(n) = 0, 3, -3, -3, 0, -15, 3, -33, 0 -57, 15, -87, 0, -123, ...
First 3 rows:
-1 0 -3 2  3  6  5 12 15 20 21 30 35
1 -3  5 1  3 -1  7  3  5  1  9  5  7
-4 8 -4 2 -4  8 -4  2 -4  8 -4  2 -4.
Note that the terms of a(n) increase from 12. Compare to increasing terms permutation of A061037(n): -3,-1,0,2,3,5,6,12,15, .... and A129647.
c(n) =  0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, ... (cf. A134967)
d(n) = -1,  1, 1,  3, 1, 3, 3, 5, 3, 5, 5, 7, 5, 7, 7, 9, ..., hence:
a(n) = c(n+1) * d(n+1).

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

[(2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018

A214297 := proc(n)
    option remember;
    if n <=5 then
        op(n+1,[-1,0,-3,2,3,6]) ;
    else
        2*procname(n-1)-procname(n-2)+procname(n-4)-2*procname(n-5)+procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

Table[(2 n^2 - 11 - 9 (-1)^n + 6 ((-1)^((2 n + 1 - (-1)^n)/4) + (-1)^((2 n - 1 + (-1)^n)/4)))/8, {n, 0, 69}] (* or *)
CoefficientList[Series[-(1 - 2 x + 4 x^2 - 8 x^3 + 3 x^4)/((1 - x)^2*(1 - x^4)), {x, 0, 69}], x] (* Michael De Vlieger, Mar 24 2017 *)

vector(100, n, n--; (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8) \\ G. C. Greubel, Sep 19 2018

a(k+4)  - a(k)    =  2*k + 4.
a(k+2)  - a(k-2)  =  2*k.
a(k+6)  - a(k-6)  =  6*k.
a(k+10) - a(k-10) = 10*k.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(2*k)   = -1, -3, followed by 3, 5, 15, 21, 35, 45, ... (A142717);
a(2*k+1) = k*(k+1) (see A002378).
A198442(n) = -1,0,0,2,3,6,8,12,  minus 3 at A198442(4*n+2).
G.f. -( 1-2*x+4*x^2-8*x^3+3*x^4 )/( (1-x)^2*(1-x^4) ). - R. J. Mathar, Jul 17 2012; edited by N. J. A. Sloane, Jul 22 2012
From R. J. Mathar, Jun 28 2013: (Start)
a(4*k)   = A000466(k);
a(4*k+1) = A002943(k);
a(4*k+2) = A078371(k-1) for k>0;
a(4*k+3) = A002939(k+1). (End)
a(n) = (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8. - Luce ETIENNE, Oct 27 2016

Edited by N. J. A. Sloane, Jul 22 2012

cp's OEIS Frontend

A125202 a(n) = 4*n^2 - 6*n + 1.

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A118729 Rectangular array where row r contains the 8 numbers 4*r^2 - 3*r, 4*r^2 - 2*r, ..., 4*r^2 + 4*r.

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A003148 a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1.

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A007622 Consider Leibniz's harmonic triangle (A003506) and look at the non-boundary terms. Sequence gives numbers appearing in denominators, sorted.

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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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A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.

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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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A125202 a(n) = 4n^2 - 6n + 1.

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

A003148 a(n+1) = a(n) + 2n(2n+1)a(n-1), with a(0) = a(1) = 1.

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.