A102130
Primes of the form 8*n^2 + 4*n + 1.
Original entry on oeis.org
13, 41, 313, 421, 1013, 1201, 1861, 2113, 2381, 3613, 5101, 7321, 9941, 10513, 13613, 14281, 16381, 20201, 21013, 21841, 24421, 30013, 34061, 41761, 47741, 51521, 52813, 54121, 59513, 60901, 82013, 83641, 90313, 97241, 99013, 100801, 106261
Offset: 1
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[ a: n in [0..400] | IsPrime(a) where a is 8*n^2 +4*n + 1]; // Vincenzo Librandi, Nov 17 2010
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Select[Table[8n^2+4n+1,{n,0,200}],PrimeQ] (* Harvey P. Dale, Jul 24 2012 *)
A103777
Numbers n such that f[n],f[n+1]and f[n+2] are all primes, where f[n]=8*n^2+4*n+1.
Original entry on oeis.org
15, 50, 80, 110, 230, 245, 425, 570, 635, 645, 710, 925, 1440, 1645, 1710, 1815, 2000, 2465, 2635, 2940, 3040, 3090, 3195, 3525, 4260, 4310, 4400, 4885, 5960, 6145, 7030, 7120, 7250, 8430, 8890, 9445, 10265, 11060, 11150, 11710, 11775, 13020, 13565
Offset: 1
15 is a term because f[15]=1861, f[16]=2113 and f[17]=2381 are all primes.
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Flatten[Position[Partition[Table[PrimeQ[8n^2+4n+1],{n,14000}],3,1],{True,True,True}]] (* Harvey P. Dale, Oct 08 2012 *)
A341470
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 41, 63, 1, 1, 9, 85, 377, 321, 1, 1, 11, 145, 1159, 3649, 1683, 1, 1, 13, 221, 2625, 16641, 36365, 8989, 1, 1, 15, 313, 4991, 50049, 246047, 369305, 48639, 1, 1, 17, 421, 8473, 118721, 982729, 3707509, 3800305, 265729, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 13, 41, 85, 145, 221, ...
1, 63, 377, 1159, 2625, 4991, ...
1, 321, 3649, 16641, 50049, 118721, ...
1, 1683, 36365, 246047, 982729, 2908411, ...
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T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));
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T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));
A343007
Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.
Original entry on oeis.org
6, 13, 26, 41, 62, 85, 114, 145, 182, 221, 266, 313, 366, 421, 482, 545, 614, 685, 762, 841, 926, 1013, 1106, 1201, 1302, 1405, 1514, 1625, 1742, 1861, 1986, 2113, 2246, 2381, 2522, 2665, 2814, 2965, 3122, 3281, 3446, 3613, 3786, 3961, 4142, 4325, 4514, 4705
Offset: 1
The first several partial sums are as follows:
n L(n)
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1 4.0000000000
2 2.6666666...
3 3.4666666...
4 2.8952380...
5 3.3396825...
6 2.9760461...
7 3.2837384...
8 3.0170718...
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For n=1, the average of the partial sums L(1) and L(2) is V = (L(1) + L(2))/2 = (4 + 2.6666666...)/2 = 3.3333333...; the two partial sums closest to V are L(5)=3.3396825... and L(7)=3.2837384..., and V lies in the interval between them, so a(1)=6.
The formula as it is written works for all data in the sequence, but it needs to be proven that it works for all possible integer values of n.
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Rest@ CoefficientList[Series[x (6 + x + x^3)/((1 + x) (1 - x)^3), {x, 0, 48}], x] (* Michael De Vlieger, Apr 05 2021 *)
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