A016922 -id:A016922 - cp's OEIS frontend

A016932 a(n) = (6*n + 1)^12.

1, 13841287201, 23298085122481, 2213314919066161, 59604644775390625, 787662783788549761, 6582952005840035281, 39959630797262576401, 191581231380566414401, 766217865410400390625, 2654348974297586158321, 8182718904632857144561, 22902048046490258711521

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929, A016930, A016931.

[(6*n+1)^12: n in [0..20]]; // Vincenzo Librandi, May 04 2011

Table[(6*n + 1)^12, {n, 0, 12}] (* Amiram Eldar, Mar 28 2022 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^12 = A016922(n)^6 = A016923(n)^4 = A016924(n)^3 = A016926(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/6)/86890185149644800. (End)

A174321 Index of the smallest prime greater than (6n+1)^2.

Original entry on oeis.org

1, 16, 40, 73, 115, 163, 220, 284, 358, 435, 520, 610, 706, 812, 924, 1039, 1164, 1295, 1424, 1573, 1716, 1878, 2033, 2191, 2367, 2548, 2730, 2916, 3108, 3303, 3513, 3732, 3946, 4165, 4397, 4628, 4858, 5107, 5357, 5612, 5883, 6148, 6415, 6685, 6961, 7253

Offset: 0

Juri-Stepan Gerasimov, Mar 15 2010

			a(0)=1 because prime(1) > (6*0 + 1)^2;
a(1)=16 because prime(16) > (6*1+1)^2 > prime(15);
a(2)=40 because prime(40) > (6*2+1)^2 > prime(39).

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Robert Israel)

Cf. A000720, A151800, A016922.

A174321 := proc(n) numtheory[pi](nextprime((6*n+1)^2)) ; end proc: seq(A174321(n),n=0..80) ; # R. J. Mathar, Mar 29 2010

A174321[n_] := PrimePi[NextPrime[(6*n + 1)^2]]; Array[A174321, 50, 0] (* Paolo Xausa, Mar 06 2025 *)

a(n) = A000720(A151800(A016922(n))). - Michel Marcus, Mar 06 2025

Definition corrected by Charles R Greathouse IV, Mar 20 2010

Offset corrected by Robert Israel, Mar 05 2025

A289134 a(n) = 21n^2 - 33n + 13.

Original entry on oeis.org

1, 31, 103, 217, 373, 571, 811, 1093, 1417, 1783, 2191, 2641, 3133, 3667, 4243, 4861, 5521, 6223, 6967, 7753, 8581, 9451, 10363, 11317, 12313, 13351, 14431, 15553, 16717, 17923, 19171, 20461, 21793, 23167, 24583, 26041, 27541, 29083, 30667, 32293

Offset: 1

Daniel Rockwitz Reynolds, Jun 25 2017

a(n) is the sum of all cells in a cellular-automata-like hexagonal lattice growth from a single active seed, based upon whether each hexagonal unit is active plus how many active neighbors each cell is touching for all active cells in the lattice.

The initial hexagonal seed starts with a single 1 representing it is active and touching no active neighbors. In the next time step, all inactive hexagonal neighboring spaces in the surrounding hexagonal lattice which were touching the active seed via edges become active and all active cells are summed together based on whether they are active plus how many active neighbors they are touching via their edges. This continues for each time step with inactive neighbors touching active neighbors in the previous time step becoming active in the current step followed by the described summing.

Colin Barker, Table of n, a(n) for n = 1..1000
D. R. Reynolds, Geometric graphic of t, a(t) for t=1...4
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033574 (analog for square tiling, von Neumann neighborhood), A016922 (analog for square tiling, Moore neighborhood), A016923 (analog for cubic 3D tiling, Moore neighborhood), A064762.

hexgro[t_]:=7+4*6+5*6*(t-2)+Sum[i*6*7,{i,t-2}]; Table[hexgro[n],{n,40}]
LinearRecurrence[{3,-3,1},{1,31,103},40] (* Harvey P. Dale, Apr 23 2020 *)

Vec(x*(1 + 28*x + 13*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jun 28 2017

G.f.: x*(1 + 28*x + 13*x^2) / (1 - x)^3. - Colin Barker, Jun 28 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - Colin Barker, Jul 29 2017

New Name from Omar E. Pol, Jun 25 2017

A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 7, 6, 4, 13, 18, 10, 5, 21, 36, 31, 15, 6, 31, 60, 64, 50, 21, 7, 43, 90, 109, 105, 71, 28, 8, 57, 126, 166, 180, 151, 98, 36, 9, 73, 168, 235, 275, 261, 210, 127, 45, 10, 91, 216, 316, 390, 401, 364, 274, 162, 55, 11, 111, 270, 409, 525, 571, 560, 477, 351, 199, 66

Offset: 1

Lamine Ngom, Sep 05 2021

A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).

In column k, every term is the arithmetic mean of its neighbors minus A000217(k).

			Array, A(n, k), begins:
  1  3   6  10  15   21   28   36   45 ... A000217;
  2  7  18  31  50   71   98  127  162 ... A195605;
  3 13  36  64 105  151  210  274  351 ...
  4 21  60 109 180  261  364  477  612 ...
  5 31  90 166 275  401  560  736  945 ...
  6 43 126 235 390  571  798 1051 1350 ...
  7 57 168 316 525  771 1078 1422 1827 ...
  8 73 216 409 680 1001 1400 1849 2376 ...
  9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
   1;
   2,  3;
   3,  7,   6;
   4, 13,  18,  10;
   5, 21,  36,  31,  15;
   6, 31,  60,  64,  50,  21;
   7, 43,  90, 109, 105,  71,  28;
   8, 57, 126, 166, 180, 151,  98,  36;
   9, 73, 168, 235, 275, 261, 210, 127,  45;
  10, 91, 216, 316, 390, 401, 364, 274, 162,  55;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Family of sequences (k*n + 1)^2: A016754 (k=2), A016778 (k=3), A016814 (k=4), A016862 (k=5), A016922 (k=6), A016994 (k=7), A017078 (k=8), A017174 (k=9), A017282 (k=10), A017402 (k=11), A017534 (k=12), A134934 (k=14).

Cf. A000027, A000217, A002061, A028896, A085473, A195605.

A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n,k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022

A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n,k-1]; Table[Function[n, A[n, k]][m-k+1], {m,0,10}, {k,0,m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)

def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # G. C. Greubel, Dec 25 2022

A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.
A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.
A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.
A(n, 0) = A000027(n).
A(n, 1) = A002061(n+1).
A(n, 2) = A028896(n).
A(n, 3) = A085473(n).
From G. C. Greubel, Dec 25 2022: (Start)
A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).
T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).
Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)

cp's OEIS Frontend

A016932 a(n) = (6*n + 1)^12.

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A174321 Index of the smallest prime greater than (6n+1)^2.

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A289134 a(n) = 21*n^2 - 33*n + 13.

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A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.

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Magma

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A289134 a(n) = 21n^2 - 33n + 13.