A194455
a(n) = 2^n + 3n + 1.
Original entry on oeis.org
2, 6, 11, 18, 29, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742, 4294967393, 8589934692, 17179869287
Offset: 0
Cf.
A062709 (first differences),
A000079 (second and successive differences).
Cf.
A146529 (differences between alternate terms, for n>2).
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[2^n+3*n+1: n in [0..31]];
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Table[2^n + 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{4,-5,2},{2,6,11},40] (* Harvey P. Dale, Oct 01 2014 *)
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for(n=0, 31, print1(2^n+3*n+1", "));
A093988
Numbers k such that 2^k + 3*k is prime.
Original entry on oeis.org
1, 3, 5, 7, 11, 13, 35, 45, 47, 57, 87, 183, 325, 367, 447, 809, 1157, 2789, 5775, 14829, 20687, 46463, 62491, 92147, 128745
Offset: 1
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A093988:=n->`if`(isprime(2^n+3*n), n, NULL): seq(A093988(n), n=1..10^3); # Wesley Ivan Hurt, Jan 21 2017
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Do[ If[ PrimeQ[2^n + 3n], Print[n]], {n, 1, 5000, 2}]
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isok(n) = isprime(2^n + 3*n); \\ Michel Marcus, Jan 21 2017
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ABC2 2^$a+3*$a
a: from 1 to 1000 // Jinyuan Wang, Feb 03 2020
A140800
Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.
Original entry on oeis.org
1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742
Offset: 0
a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).
- H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
- Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
- P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.
Cf.
A000943,
A000944,
A019503,
A053016,
A060296,
A063924,
A063925,
A063926,
A063927,
A065984,
A086653,
A093478,
A093479,
A105230,
A105231.
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LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)
Showing 1-3 of 3 results.
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