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Integers n such that A005897(n) is the sum of two positive cubes.
+20
0
5, 11, 17, 28, 37, 81, 87, 107, 141, 178, 200, 205, 229, 296, 301, 377, 385, 395, 427, 497, 511, 595, 613, 641, 660, 907, 914, 921, 955, 975, 983, 991, 1043, 1129, 1265, 1343, 1369, 1382, 1409, 1537, 1552, 1601, 1819, 1838, 1839, 1917, 1922, 1979, 2205, 2299, 2381, 2581, 2649, 2663
COMMENTS
The number of unit cubes on the surface of an n X n X n cube is given by sequence A005897. With a pair of scales, one might incorrectly think a single cube could be equal to the sum of two cubes. However, we know this is impossible because of Fermat's Last Theorem.
But we can put a 6 X 6 X 6 cube containing only surface unit cubes on one scale: there are 152 unit cubes. In other side of the scale we can put a 3 X 3 X 3 cube and a 5 X 5 X 5 cube, so there are 27 unit cubes and 125 unit cubes, and the two pans balance.
EXAMPLE
5 is a term because A005897(5) = 152 = 3^3 + 5^3. 11 is a term because A005897(11) = 728 = 6^3 + 8^3. 17 is a term because A005897(17) = 1736 = 2^3 + 12^3.
MATHEMATICA
Select[Range@ 2700, Length[PowersRepresentations[6 #^2 + 2, 2, 3] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Feb 01 2016 *)
PROG
(PARI) T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
a(n) = if(n, 6*n^2+2, 1);
for(n=0, 1e4, if(is(a(n)), print1(n, ", ")));
Integers n such that A005897(n) is the sum of two nonzero squares.
+20
0
1, 2, 4, 6, 7, 8, 9, 12, 15, 16, 20, 22, 23, 26, 28, 30, 34, 36, 40, 41, 44, 47, 49, 50, 54, 55, 56, 57, 58, 63, 64, 65, 68, 70, 78, 82, 84, 86, 89, 94, 97, 98, 102, 103, 104, 105, 106, 111, 112, 113, 118, 120, 126, 127, 132, 135, 142, 143, 145, 146, 148, 151, 152, 153, 154, 156
COMMENTS
Integers n such that number of points on surface of a 3-dimensional cube in which each face has a square grid of dots drawn on it (with n+1 points along each edge, including the corners) is the sum of two nonzero squares.
EXAMPLE
1 is a term because A005897(1) = 8 = 2^2 + 2^2. 2 is a term because A005897(2) = 26 = 1^2 + 5^2. 4 is a term because A005897(4) = 98 = 7^2 + 7^2. 6 is a term because A005897(6) = 218 = 7^2 + 13^2.
PROG
(PARI) is_a000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
a005897(n) = if(n, 6*n^2+2, 1);
for(n=0, 200, if(is_a000404(a005897(n)), print1(n, ", ")));
1, 4, 26, 56, 98, 152, 218, 296, 386, 488, 602, 728, 866, 1016, 1178, 1352, 1538, 1736, 1946, 2168, 2402, 2648, 2906, 3176, 3458, 3752, 4058, 4376, 4706, 5048, 5402, 5768, 6146, 6536, 6938, 7352, 7778, 8216, 8666, 9128, 9602, 10088
FORMULA
a(n) = 6*n^2 - 12*n + 8 = A005897(n-1) for n > 2. a(n) = n^3 - (n-2)^3 for n > 2. - Joerg Arndt, Jun 06 2016
Coordination sequence T1 for Zeolite Code ABW and ATN.
+10
793
1, 4, 10, 21, 36, 54, 78, 106, 136, 173, 214, 256, 306, 360, 414, 477, 544, 610, 686, 766, 844, 933, 1026, 1116, 1218, 1324, 1426, 1541, 1660, 1774, 1902, 2034, 2160, 2301, 2446, 2584, 2738, 2896, 3046, 3213, 3384, 3546, 3726, 3910, 4084, 4277, 4474, 4660
REFERENCES
W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996
FORMULA
a(3m)=19m^2+2, a(3m+1)=19m^2+13m+4, a(3m+2)=19m^2+25m+10, for m>0. [ N. J. A. Sloane]
MATHEMATICA
CoefficientList[Series[(-z^7 - 3 z^6 - 6 z^5 - 9 z^4 - 9 z^3 - 6 z^2 - 3 z - 1)/((z - 1)^3 (z^2 + z + 1)^2), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
PROG
(PARI) a(n)=if(n, my(m=divrem(n, 3)); 19*m[1]^2+if(m[2], if(m[2]==1, 13*m[1]+4, 25*m[1]+10), 2), 1) \\ Charles R Greathouse IV, Jun 28 2011 (Magma) I:=[1, 4, 10, 21, 36, 54, 78, 106]; [n le 8 select I[n] else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Jun 10 2013
Centered cube numbers: n^3 + (n+1)^3.
(Formerly M4616)
+10
112
1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, 46341, 51389, 56791, 62559, 68705, 75241, 82179, 89531, 97309, 105525, 114191, 123319, 132921
COMMENTS
Write the natural numbers in groups: 1; 2,3,4; 5,6,7,8,9; 10,11,12,13,14,15,16; ..... and add the groups, i.e., a(n) = Sum_{j=n^2-2(n-1)..n^2} j. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 05 2001
The numbers 1, 9, 35, 91, etc. are divisible by 1, 3, 5, 7, etc. Therefore there are no prime numbers in this list. 9 is divisible by 3 and every third number after 9 is also divisible by 3. 35 is divisible by 5 and 7 and every fifth number after 35 is also divisible by 5 and every seventh number after 35 is also divisible by 7. This pattern continues indefinitely. - Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008
n^3 + (n+1)^3 = (2n+1)*(n^2+n+1), hence all terms are composite. - Zak Seidov, Feb 08 2011 This is the order of an n-ball centered at a node in the Kronecker product (or direct product) of three cycles, each of whose lengths is at least 2n+2. - Pranava K. Jha, Oct 10 2011 Positive y values of 4*x^3 - 3*x^2 = y^2. - Bruno Berselli, Apr 28 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
FORMULA
E.g.f.: (1 + 8*x + 9*x^2 + 2*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
MATHEMATICA
CoefficientList[Series[(1 + 5 x + 5 x^2 + x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 16 2015 *)
PROG
(Python)
A005898_list, m = [], [12, -6, 2, 1]
for _ in range(10**2):
for i in range(3):
CROSSREFS
(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Centered octahedral numbers (crystal ball sequence for cubic lattice).
(Formerly M4384 N1844)
+10
93
1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919, 11521, 13287, 15225, 17343, 19649, 22151, 24857, 27775, 30913, 34279, 37881, 41727, 45825, 50183, 54809, 59711, 64897, 70375, 76153, 82239
COMMENTS
Number of points in simple cubic lattice at most n steps from origin.
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 6-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007 Equals binomial transform of [1, 6, 12, 8, 0, 0, 0, ...] where (1, 6, 12, 8) = row 3 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 4, a(n-2) = -coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 26 2010 a(n) = D(3,n) where D are the Delannoy numbers ( A008288). As such, a(n) gives the number of grid paths from (0,0) to (3,n) using steps that move one unit north, east, or northeast. - David Eppstein, Sep 07 2014 The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^3 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array ( A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 3 from any given point. - Shel Kaphan, Jan 02 2023
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy) T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
FORMULA
G.f.: (1+x)^3 /(1-x)^4. [conjectured (correctly) by Simon Plouffe in his 1992 dissertation] a(n) = (2*n+1)*(2*n^2 + 2*n + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=7, a(2)=25, a(3)=63. - Harvey P. Dale, Jun 05 2013 a(n) = Sum_{k=0..min(3,n)} 2^k * binomial(3,k) * binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014 E.g.f.: exp(x)*(3 + 18*x + 18*x^2 + 4*x^3)/3. - Stefano Spezia, Mar 14 2024 Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 5/6 - log(2) = (1 - 1/2 + 1/3) - log(2). - Peter Bala, Mar 21 2024
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {1, 7, 25, 63}, 40] (* Harvey P. Dale, Jun 05 2013 *) CoefficientList[Series[(1 + x)^3/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)
PROG
(Haskell)
a001845 n = (2 * n + 1) * (2 * n ^ 2 + 2 * n + 3) `div` 3
CROSSREFS
Sums of 2 consecutive terms give A008412. (1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.
(Formerly M4898)
+10
86
1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, 69355, 77197, 85609, 94611, 104223, 114465, 125357, 136919, 149171, 162133, 175825, 190267, 205479
COMMENTS
Called "magic numbers" in some chemical contexts.
Equals binomial transform of [1, 12, 30, 20, 0, 0, 0, ...]. - Gary W. Adamson, Aug 01 2008 Crystal ball sequence for A_3 lattice. - Michael Somos, Jun 03 2012
REFERENCES
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf). D. R. Herrick, Home Page (displays these numbers as sizes of clusters in chemistry) T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
FORMULA
a(n) = (2*n+1)*(5*n^2+5*n+3)/3.
G.f.: (x^3 + 9x^2 + 9x + 1)/(x - 1)^4.
E.g.f.: (1/3)*exp(x)*(10x^3 + 45x^2 + 36x + 3).
(End)
EXAMPLE
a(4) = 147 = (1, 3, 3, 1) dot (1, 12, 30, 20) = (1 + 36 + 90 + 20). - Gary W. Adamson, Aug 01 2008 G.f. = 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 + ...
MAPLE
A005902 := n -> (2*n+1)*(5*n^2+5*n+3)/3;
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {1, 13, 55, 147}, 50] (* Harvey P. Dale, Oct 08 2015 *) CoefficientList[Series[(x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4, {x, 0, 50}], x] (* Indranil Ghosh, Apr 08 2017 *)
PROG
(PARI) {a(n) = (2*n + 1) * (5*n^2 + 5*n + 3) / 3}; /* Michael Somos, Jun 03 2012 */ (PARI) x='x+O('x^50); Vec((x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4) \\ Indranil Ghosh, Apr 08 2017 (Magma) [(2*n+1)*(5*n^2+5*n+3)/3: n in [0..30]]; // G. C. Greubel, Dec 01 2017 (Python)
def a(n): return (2*n+1)*(5*n**2+5*n+3)//3
CROSSREFS
(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).
(Formerly M3380)
+10
84
1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234
COMMENTS
Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - Emeric Deutsch, Dec 25 2004 For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk, May 20 2006; updated by Peter Munn, Aug 25 2017 due to changed offset in A000292Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008 Disregarding the terms < 10, the sums of four consecutive triangular numbers ( A000217). - Rick L. Shepherd, Sep 30 2009 Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - Frank M Jackson, Sep 07 2011 Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014 Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016 For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - Muniru A Asiru, Apr 19 2016
REFERENCES
N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf). Reticular Chemistry Structure Resource, sod.
FORMULA
G.f.: (1 - x^4)/(1-x)^4.
a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008 a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 26 2012 For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016 Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)
Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - Carlo Wood, Dec 04 2023 Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - Charlie Marion, May 15 2024
EXAMPLE
G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...
MATHEMATICA
Join[{1}, LinearRecurrence[{3, -3, 1}, {4, 10, 20}, 50]] (* Harvey P. Dale, Feb 26 2012 *) a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *) a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)
CROSSREFS
Cf. similar sequences listed in A255843. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.
(Formerly M4115)
+10
75
1, 6, 18, 38, 66, 102, 146, 198, 258, 326, 402, 486, 578, 678, 786, 902, 1026, 1158, 1298, 1446, 1602, 1766, 1938, 2118, 2306, 2502, 2706, 2918, 3138, 3366, 3602, 3846, 4098, 4358, 4626, 4902, 5186, 5478, 5778, 6086, 6402, 6726, 7058, 7398, 7746, 8102, 8466
COMMENTS
Also, the number of regions the plane can be cut into by two overlapping concave (2n)-gons. - Joshua Zucker, Nov 05 2002 If X is an n-set and Y_i (i=1,2,3) are mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007 Also the least number of unit cubes required, at the n-th iteration, to surround a 3D solid built from unit cubes, in order to hide all its visible faces, starting with a unit cube. - R. J. Cano, Sep 29 2015 Also, coordination sequence for "tfs" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018 Also, the number of n-th order specular reflections arriving at a receiver point from an emitter point inside a cuboid with reflective faces. - Michael Schutte, Sep 18 2018
REFERENCES
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF8
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #16 and #22.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
Binomial transform of [1, 5, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 02 2007 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=1, a(1)=6, a(2)=18, a(3)=38. - Harvey P. Dale, Nov 08 2011 Recurrence: n*a(n) = (n-2)*a(n-2) + 6*a(n-1), a(0)=1, a(1)=6. - Fung Lam, Apr 15 2014 a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - Shel Kaphan, Feb 16 2023 Sum_{n>=0} 1/a(n) = 3/4 + Pi *sqrt(2)*coth( Pi/sqrt 2)/8 = 1.31858... - R. J. Mathar, Apr 27 2024
MATHEMATICA
Join[{1}, 4Range[40]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {6, 18, 38}, 40]] (* Harvey P. Dale, Nov 08 2011 *)
PROG
(PARI) Vec(((1+x)/(1-x))^3 + O(x^100)) \\ Altug Alkan, Oct 26 2015
CROSSREFS
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Centered tetrahedral numbers.
(Formerly M3850)
+10
71
1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115
COMMENTS
Binomial transform of (1,4,6,4,0,0,0,...). - Paul Barry, Jul 01 2003 If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = (2*n + 1)*(n^2 + n + 3)/3.
G.f.: (1+x)*(1+x^2)/(1-x)^4.
a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). - Paul Barry, Jul 01 2003 a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified by G. C. Greubel, Nov 30 2017) a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. - Harvey P. Dale, Nov 03 2011 E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. - G. C. Greubel, Nov 30 2017
MATHEMATICA
Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 5, 15, 35}, 40] (* Harvey P. Dale, Nov 03 2011 *)
PROG
(Magma) [(2*n+1)*(n^2+n+3)/3: n in [0..30]]; // G. C. Greubel, Nov 30 2017
CROSSREFS
(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
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