A010673 -id:A010673 - cp's OEIS frontend

A248825 a(n) = n^2 + 1 - (-1)^n.

0, 3, 4, 11, 16, 27, 36, 51, 64, 83, 100, 123, 144, 171, 196, 227, 256, 291, 324, 363, 400, 443, 484, 531, 576, 627, 676, 731, 784, 843, 900, 963, 1024, 1091, 1156, 1227, 1296, 1371, 1444, 1523, 1600, 1683, 1764, 1851, 1936, 2027, 2116

Offset: 0

Paul Curtz, Oct 15 2014

Also, A016742 and A164897 interleaved.
See the spiral in Example field of A054552: after 0, the sequence is given by the terms of the semidiagonals 4, 16, 36, 64, 100, ... and 3, 11, 27, 51, 83, ... sorted into ascending order.
Primes of the sequence are in A056899.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290, A008586, A008602, A010673, A016742, A042963, A056899, A164897, A166519, A168277, A248800.

[n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014

Table[n^2 + 1 - (-1)^n, {n, 0, 60}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,4,11},60] (* Harvey P. Dale, Jun 30 2019 *)

vector(100,n,(n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014

[n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

a(n) = a(-n) = 2*a(n-1) - 2*(n-3) + a(n-4).
a(n) = n^2 + A010673(n) = (n+1)^2 - A168277(n+1).
a(n+1) = A248800(n) + A042963(n+1) = a(n) + A166519(n).
a(n+2) = a(n) + 4*n.
a(n+5) = a(n-5) + A008602(n).
G.f.: x*(3 - 2*x + 3*x^2)/((1 + x)*(1 - x)^3). - Bruno Berselli, Oct 15 2014
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022

Edited by Bruno Berselli, Oct 16 2014

A277443 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the prism graph Y_k on 2k vertices.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 18, 0, 0, 2, 12, 84, 0, 0, 0, 114, 264, 260, 0, 0, 2, 180, 2652, 1920, 630, 0, 0, 0, 858, 16080, 29660, 8520, 1302, 0, 0, 2, 1932, 119844, 367080, 198030, 28140, 2408, 0, 0, 0, 7074, 816984, 4843460, 4067280, 932862, 76272, 4104, 0, 0, 2, 18660, 5784492, 62682480, 85847910, 28576380, 3440024, 179424, 6570, 0

Offset: 1

Jeremy Tan, Oct 15 2016

Y_1 contains a loop, so has no colorings with any number of colors. Y_2 is the cycle graph C_4 with two double edges; these two graphs are therefore equivalent with respect to number of colorings.

			Square array A(n,k) begins:
  0,   0,    0,      0,       0,        0,          0, ...
  0,   2,    0,      2,       0,        2,          0, ...
  0,  18,   12,    114,     180,      858,       1932, ...
  0,  84,  264,   2652,   16080,   119844,     816984, ...
  0, 260, 1920,  29660,  367080,  4843460,   62682480, ...
  0, 630, 8520, 198030, 4067280, 85847910, 1800687000, ...

N. L. Biggs, R. M. Damerell and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131. MR0294172
Eric Weisstein's World of Mathematics, Prism Graph
Wikipedia, Chromatic polynomial

Cf. A277444 (colorings of Möbius ladders), A182406 (square grid graphs).
Columns k=1,2 are A000004 and A091940.
Rows n=1,2 are A000004 and A010673.

A(n,k) = (n^2-3n+3)^k+(n-1)((3-n)^k+(1-n)^k)+n^2-3n+1.

A277444 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the Möbius ladder M_k on 2k vertices.

Original entry on oeis.org

0, 0, 2, 0, 0, 6, 0, 2, 0, 12, 0, 0, 42, 24, 20, 0, 2, 48, 420, 120, 30, 0, 0, 306, 2160, 2420, 360, 42, 0, 2, 600, 17532, 27600, 9750, 840, 56, 0, 0, 2442, 115464, 375260, 191760, 30702, 1680, 72, 0, 2, 6048, 830100, 4810680, 4098510, 917280, 81032, 3024, 90, 0, 0, 20706, 5745120, 62813540, 85691640, 28669662, 3406368, 187560, 5040, 110

Offset: 1

Jeremy Tan, Oct 15 2016

M_1 is two vertices connected by a triple edge and thus behaves like the path graph P_2 in terms of colorings. M_2 is isomorphic to K_4, the tetrahedral graph.

			Square array A(n,k) begins:
0,    0,    0,      0,       0,        0,          0, ...
2,    0,    2,      0,       2,        0,          2, ...
6,    0,   42,     48,     306,      600,       2442, ...
12,  24,  420,   2160,   17532,   115464,     830100, ...
20, 120, 2420,  27600,  375260,  4810680,   62813540, ...
30, 360, 9750, 191760, 4098510, 85691640, 1801468230, ...

N. L. Biggs, R. M. Damerell and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131. MR0294172
Eric Weisstein's World of Mathematics, Möbius Ladder
Wikipedia, Chromatic polynomial

Cf. A277443 (colorings of prism graphs), A182406 (square grid graphs).

Columns k=1,2 are A002378 and A052762. Rows n=1,2 are A000004 and A010673.

A(n,k) = (n^2-3n+3)^k+(n-1)((3-n)^k-(1-n)^k)-1.

cp's OEIS Frontend

A248825 a(n) = n^2 + 1 - (-1)^n.

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A277443 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the prism graph Y_k on 2k vertices.

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A277444 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the Möbius ladder M_k on 2k vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula