A033996 -id:A033996 - cp's OEIS frontend

A124080 10 times triangular numbers: a(n) = 5n(n + 1).

0, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550, 660, 780, 910, 1050, 1200, 1360, 1530, 1710, 1900, 2100, 2310, 2530, 2760, 3000, 3250, 3510, 3780, 4060, 4350, 4650, 4960, 5280, 5610, 5950, 6300, 6660, 7030, 7410, 7800, 8200, 8610, 9030, 9460, 9900, 10350

Offset: 0

Zerinvary Lajos, Nov 24 2006

If Y is a 5-subset of an n-set X then, for n >= 5, a(n-4) is equal to the number of 5-subsets of X having exactly three elements in common with Y. Y is a 5-subset of an n-set X then, for n >= 6, a(n-6) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007

Also sequence found by reading the line from 0, in the direction 0, 10, ... and the same line from 0, in the direction 0, 30, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Axis perpendicular to A195148 in the same spiral. - Omar E. Pol, Sep 18 2011

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

Cf. A028895, A046092, A045943, A002378, A028896, A024966, A033996, A027468, A049598, A000217.

[ 5*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

[seq(10*binomial(n,2),n=1..51)];
seq(n*(n+1)*5, n=0..39); # Zerinvary Lajos, Mar 06 2007

10*Accumulate[Range[0,50]] (* or *) LinearRecurrence[{3,-3,1},{0,10,30},50] (* Harvey P. Dale, Jul 21 2011 *)

a(n)=5*n*(n+1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 10*C(n,2), n >= 1.
a(n) = A049598(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007
a(n) = 5*n*(n + 1), n >= 0. - Zerinvary Lajos, Mar 06 2007
a(n) = 5*n^2 + 5*n = 10*A000217(n) = 5*A002378(n) = 2*A028895(n). - Omar E. Pol, Dec 12 2008
a(n) = 10*n + a(n-1) (with a(0) = 0). - Vincenzo Librandi, Nov 12 2009
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 10, a(2) = 30. - Harvey P. Dale, Jul 21 2011
a(n) = A062786(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A131242(10*n+9). - Philippe Deléham, Mar 27 2013
From G. C. Greubel, Aug 22 2017: (Start)
G.f.: 10*x/(1 - x)^3.
E.g.f.: 5*x*(x + 2)*exp(x). (End)
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2)-1)/5. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(5/Pi)*cos(3*Pi/(2*sqrt(5))).
Product_{n>=1} (1 + 1/a(n)) = (5/Pi)*cos(Pi/(2*sqrt(5))). (End)

A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

Original entry on oeis.org

83, 101, 127, 137, 163, 199, 233, 311, 373, 443, 463, 491, 541, 587, 613, 631, 641, 659, 673, 683, 691, 733, 757, 797, 859, 881, 911, 919, 953, 971, 991, 1013, 1051, 1061, 1103, 1109, 1117, 1193, 1201, 1213, 1249, 1307, 1319, 1409, 1433, 1459, 1483, 1487

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006

nonn

Isolated prime numbers have no adjacent primes in a lattice generated by writing consecutive integers starting from 1 in a spiral distribution. If n0 is the number of isolated primes and p the number of primes less than N, the ratio n0/p approaches 1 as N increases. If n1, n2, n3, n4 denote the number of primes with respectively 1, 2, 3, 4 adjacent primes in the lattice, the ratios n1/n0, n2/n1, n3/n2, n4/n3 approach 0 as N increases. The limits stand for any 2D lattice of integers generated by a priori criteria (i.e., not knowing distributions of primes) as Ulam's lattice.

			83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 17^2 showing only primes, with the isolated primes in parentheses (redrawn by _Jon E. Schoenfield_, Aug 06 2017):
  257 .  .  .  .  . 251 .  .  .  .  .  .  .  .  . 241
   . 197 .  .  . 193 . 191 .  .  .  .  .  .  .  .  .
   .  .  .  .  .  .  .  . 139 .(137).  .  .  .  . 239
   .(199).(101).  .  . 97  .  .  .  .  .  .  . 181 .
   .  .  .  .  .  .  .  . 61  . 59  .  .  . 131 .  .
   .  .  . 103 . 37  .  .  .  .  . 31  . 89  . 179 .
  263 . 149 . 67  . 17  .  .  . 13  .  .  .  .  .  .
   .  .  .  .  .  .  .  5  .  3  . 29  .  .  .  .  .
   .  . 151 .  .  . 19  .  .  2 11  . 53  .(127).(233)
   .  .  . 107 . 41  .  7  .  .  .  .  .  .  .  .  .
   .  .  .  . 71  .  .  . 23  .  .  .  .  .  .  .  .
   .  .  . 109 . 43  .  .  . 47  .  .  .(83) . 173 .
  269 .  .  . 73  .  .  .  .  . 79  .  .  .  .  . 229
   .  .  .  .  . 113 .  .  .  .  .  .  .  .  .  .  .
  271 . 157 .  .  .  .  .(163).  .  . 167 .  .  . 227
   . 211 .  .  .  .  .  .  .  .  .  .  . 223 .  .  .
   .  .  .  . 277 .  .  . 281 . 283 .  .  .  .  .  .
(End)

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.

Cf. A113688 (isolated semiprimes in the semiprime spiral), A156859.

# A is Ulam's lattice
if (isprime(A[x,y])and(not(isprime(A[x+1,y]) or isprime(A[x-1,y])or isprime(A[x,y+1])or isprime(A[x,y-1])or isprime(A[x-1,y-1])or isprime(A[x+1,y+1])or isprime(A[x+1,y-1])or isprime(A[x-1,y+1])))) then print (A[x,y]) ; fi;

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@ # &]]; f[spiral@ 21 /. n_ /; CompositeQ@ n -> 0] (* Michael De Vlieger, Dec 22 2015, Version 10 *)

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

A195146 Concentric 16-gonal numbers.

Original entry on oeis.org

0, 1, 16, 33, 64, 97, 144, 193, 256, 321, 400, 481, 576, 673, 784, 897, 1024, 1153, 1296, 1441, 1600, 1761, 1936, 2113, 2304, 2497, 2704, 2913, 3136, 3361, 3600, 3841, 4096, 4353, 4624, 4897, 5184, 5473, 5776, 6081, 6400, 6721, 7056, 7393, 7744, 8097, 8464

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric hexadecagonal numbers or concentric hexakaidecagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 16, ..., and the same line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Main axis, perpendicular to A033996 in the same spiral.

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

A016802 and A195315 interleaved.
Cf. A032528, A033996, A074377, A077221, A195142, A195143, A195145, A195147, A195148, A195149.
Column 16 of A195040.

[(8*n^2+3*(-1)^n-3)/2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 16, 33}, 50] (* Amiram Eldar, Jan 16 2023 *)

a(n)=(8*n^2+3*(-1)^n-3)/2 \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = (8*n^2 + 3*(-1)^n - 3)/2;
a(n) = -a(n-1) + 8*n^2 - 8*n + 1. (End)
G.f. -x*(1+14*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi^2/96 + tan(sqrt(3)*Pi/4)*Pi/(8*sqrt(3)). - Amiram Eldar, Jan 16 2023

A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).

Original entry on oeis.org

0, 8, 28, 60, 104, 160, 228, 308, 400, 504, 620, 748, 888, 1040, 1204, 1380, 1568, 1768, 1980, 2204, 2440, 2688, 2948, 3220, 3504, 3800, 4108, 4428, 4760, 5104, 5460, 5828, 6208, 6600, 7004, 7420, 7848, 8288, 8740, 9204, 9680, 10168, 10668, 11180, 11704, 12240

Offset: 0

N. J. A. Sloane

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
Sequence found by reading the line from 0, in the direction 0, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - Omar E. Pol, Sep 09 2011
a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch May 13 2018
The partial sums of this sequence give A035006. - Leo Tavares, Oct 03 2021

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10 (2016), pp. 884-892.
Leo Tavares, Illustration: Crossed Stars
Leo Tavares, Illustration: Four Quarter Star Crosses
Leo Tavares, Illustration: Triangulated Star Crosses
Leo Tavares, Illustration: Oblong Star Crosses
Leo Tavares, Illustration: Crossed Diamond Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033579, A049451, A045945, A186423.
Cf. sequences listed in A254963.
Cf. A003154, A016813.
Cf. A035006.
Cf. A005449, A046092, A033996, A002378, A016742, A000290, A016754, A001105.

List([0..50], n-> 2*n*(3*n+1)); # G. C. Greubel, Oct 09 2019

[2*n*(3*n+1): n in [0..50]]; // G. C. Greubel, Oct 09 2019

seq(2*n*(3*n+1), n=0..50); # G. C. Greubel, Oct 09 2019

4*Binomial[3*Range[50]-2, 2]/3 (* G. C. Greubel, Oct 09 2019 *)

a(n)=2*n*(3*n+1) \\ Charles R Greathouse IV, Sep 28 2015

[2*n*(3*n+1) for n in (0..50)] # G. C. Greubel, Oct 09 2019

a(n) = a(n-1) +12*n -4 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010
G.f.: 4*x*(2+x)/(1-x)^3. - Colin Barker, Feb 13 2012
a(-n) = A033579(n). - Michael Somos, Jun 09 2014
E.g.f.: 2*x*(4 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 3/2 - Pi/(4*sqrt(3)) - 3*log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) =  -3/2 + Pi/(2*sqrt(3)) + log(2). (End)
From Leo Tavares, Oct 12 2021: (Start)
a(n) = A003154(n+1) - A016813(n). See Crossed Stars illustration.
a(n) = 4*A005449(n). See Four Quarter Star Crosses illustration.
a(n) = 2*A049451(n).
a(n) = A046092(n-1) + A033996(n). See Triangulated Star Crosses illustration.
a(n) = 4*A000217(n-1) + 8*A000217(n).
a(n) = 4*A000217(n-1) + 4*A002378. See Oblong Star Crosses illustration.
a(n) = A016754(n) + 4*A000217(n). See Crossed Diamond Stars illustration.
a(n) = 2*A001105(n) + 4*A000217(n).
a(n) = A016742(n) + A046092(n).
a(n) = 4*A000290(n) + 4*A000217(n). (End)

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

A200165 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with nonzero sum and with zero through n-1 differences all nonzero.

Original entry on oeis.org

2, 4, 0, 6, 8, 2, 8, 24, 34, 0, 10, 48, 128, 76, 2, 12, 80, 348, 576, 276, 0, 14, 120, 726, 2256, 2778, 700, 2, 16, 168, 1326, 6160, 15040, 12440, 2276, 0, 18, 224, 2180, 13888, 52486, 96884, 57992, 5992, 2, 20, 288, 3352, 27160, 145482, 438968, 631638, 262444

Offset: 1

R. H. Hardin Nov 13 2011

Table starts
.2.....4.......6.........8.........10.........12........14.........16........18
.0.....8......24........48.........80........120.......168........224.......288
.2....34.....128.......348........726.......1326......2180.......3352......4874
.0....76.....576......2256.......6160......13888.....27160......48380.....80096
.2...276....2778.....15040......52486.....145482....336992.....695778...1309052
.0...700...12440.....96884.....438968....1504820...4141212....9922424..21253088
.2..2276...57992....631638....3687480...15586972..50917674..141509152.344941886
.0..5992..262444...4072228...30776992..160772064.624074768.2012800872
.2.18542.1202540..26327538..257108784.1658521710
.0.50488.5463896.169703920.2144437176

			Some solutions for n=6 k=5
..1...-4....1....1....1...-5....1....1....2...-5...-4....2....2....2...-4....1
.-3....2....4...-5...-4....1....2...-3....1....2...-5...-5....3....3...-5....2
.-2...-1....1...-4....5...-3....4...-5...-4...-3...-3....2....1....2....1...-5
..5...-3...-3...-2...-3....5...-2....3....2...-1...-5....3...-3....3....3...-1
.-2....5....2....2...-5....4...-5...-1...-2....5...-4...-1....5....2...-1...-3
..3....4....5...-3...-2...-1....1....4...-1...-1....1...-2....2...-3....2...-4

R. H. Hardin, Table of n, a(n) for n = 1..113

Row 2 is A033996(n-1)

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

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

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

cp's OEIS Frontend

A124080 10 times triangular numbers: a(n) = 5*n*(n + 1).

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A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

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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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A195146 Concentric 16-gonal numbers.

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A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).

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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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A200165 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with nonzero sum and with zero through n-1 differences all nonzero.

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A124080 10 times triangular numbers: a(n) = 5n(n + 1).

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

A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).