A000969 -id:A000969 - cp's OEIS frontend

A000970 Fermat coefficients.

1, 7, 25, 66, 143, 273, 476, 775, 1197, 1771, 2530, 3510, 4750, 6293, 8184, 10472, 13209, 16450, 20254, 24682, 29799, 35673, 42375, 49980, 58565, 68211, 79002, 91025, 104371, 119133, 135408, 153296, 172900, 194327, 217686, 243090, 270655

Offset: 5

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A258708.

a000970 n = a258708 n (n - 5)  -- Reinhard Zumkeller, Jun 23 2015

A000970:=-(2*z**4+3*z**5+3*z**2+4*z**3+3*z+1)/(z**4+z**3+z**2+z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(3x^5+2x^4+4x^3+3x^2+3x+1)/((1-x^5)(1-x)^4),{x,0,50}],x] (* Vincenzo Librandi, Mar 28 2012 *)
LinearRecurrence[{4,-6,4,-1,1,-4,6,-4,1},{1,7,25,66,143,273,476,775,1197},40] (* Harvey P. Dale, Sep 06 2017 *)

Vec((3*x^5+2*x^4+4*x^3+3*x^2+3*x+1)/(1-x^5)/(1-x)^4+O(x^99)) \\ Charles R Greathouse IV, Mar 28 2012

G.f.: x^5(3x^5 + 2x^4 + 4x^3 + 3x^2 + 3x + 1)/((1-x^5)(1-x)^4).

a(n) = A258708(n,n-5) = A258708(2*n-7,2). - Reinhard Zumkeller, Jun 23 2015

More terms from Sean A. Irvine, Sep 25 2011

A000971 Fermat coefficients.

Original entry on oeis.org

1, 9, 42, 132, 334, 728, 1428, 2584, 4389, 7084, 10963, 16380, 23751, 33563, 46376, 62832, 83657, 109668, 141778, 181001, 228459, 285384, 353127, 433160, 527085, 636636, 763686, 910252, 1078500, 1270752, 1489488, 1737355, 2017169, 2331924

Offset: 6

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
Index entries for linear recurrences with constant coefficients, signature (6,-15,19,-9,-9,18,-9,-9,19,-15,6,-1).

Cf. A258708.

a000971 n = a258708 n (n - 6)  -- Reinhard Zumkeller, Jun 23 2015

(1+3*z+3*z^7+z^8+3*z^2-4*z^3+10*z^4-4*z^5+3*z^6)/(z^6+z^3+1)/(-1+z)^6;

CoefficientList[Series[(1+3*x+3*x^7+x^8+3*x^2-4*x^3+10*x^4-4*x^5+3*x^6)/(x^6+x^3+1)/(-1+x)^6,{x,0,40}],x] (* Vincenzo Librandi, Mar 28 2012 *)

Vec((1+3*z+3*z^7+z^8+3*z^2-4*z^3+10*z^4-4*z^5+3*z^6)/(z^6+z^3+1)/(z-1)^6+O(x^99)) \\ Charles R Greathouse IV, Mar 28 2012

G.f.: (1 + 3x + 3x^7 + x^8 + 3x^2 - 4x^3 + 10x^4 - 4x^5 + 3x^6)/(x^6 + x^3 + 1)/(-1+x)^6 (see MAPLE line).

a(n) = A258708(n,n-6). - Reinhard Zumkeller, Jun 23 2015

More terms from Sean A. Irvine, Sep 25 2011

A000972 Fermat coefficients.

Original entry on oeis.org

1, 12, 66, 245, 715, 1768, 3876, 7752, 14421, 25300, 42287, 67860, 105183, 158224, 231880, 332112, 466089, 642341, 870922, 1163580, 1533939, 1997688, 2572780, 3279640, 4141382, 5184036, 6436782, 7932196, 9706503, 11799840, 14256528, 17125353, 20459857

Offset: 7

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 7..1000
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Cf. A258708.

a000972 n = a258708 n (n - 7)  -- Reinhard Zumkeller, Jun 23 2015

a := n->floor((2*n)*(2*n+1)*(2*n+2)*(2*n+3)*(2*n+4)*(2*n+5)/7!);

Table[Floor[((2*n)*(2*n+1)*(2*n+2)*(2*n+3)*(2*n+4)*(2*n+5)/7!)],{n,1,30}] (* Vincenzo Librandi, Apr 10 2012 *)
With[{c=7!,t=Times@@(2n+Range[0,5])},Table[Floor[t/c],{n,30}]] (* Harvey P. Dale, Apr 20 2014 *)

Vec(x^7*(1 + 6*x + 9*x^2 + 9*x^3 + 10*x^4 + 7*x^5 + 12*x^6 + 6*x^7 + 4*x^8) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^50)) \\ Colin Barker, Mar 28 2017

a(n) = A258708(n,n-7). - Reinhard Zumkeller, Jun 23 2015

G.f.: x^7*(1 + 6*x + 9*x^2 + 9*x^3 + 10*x^4 + 7*x^5 + 12*x^6 + 6*x^7 + 4*x^8) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, Mar 28 2017

A011793 Triangle of numbers of irreducible Euler sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 3, 1, 2, 5, 3, 1, 1, 5, 7, 4, 1, 3, 8, 9, 4, 1, 1, 7, 14, 12, 5, 1, 3, 14, 20, 15, 5, 1, 1, 9, 25, 30, 18, 6, 1, 4, 20, 42, 40, 22, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1

Offset: 1

N. J. A. Sloane, David Broadhurst

Vincenzo Librandi, Rows n = 1..100, flattened
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980. [Taking every other row of this triangle gives the triangle (A258708) in Table 3.]

Cf. A258708.

t[n_, k_] := (2/(n+k))*Sum[ If[ EvenQ[d], MoebiusMu[d/2]*Binomial[(n+k)/d, (n-k)/d], 0], {d, Intersection[ Divisors[n+k], Divisors[n-k]]}]; t[1, 1] = t[2, 1] = 1;row[1] = row[2] = {1}; row[n_] := Table[t[n, k], {k, 2 - Mod[n, 2], n-1, 2}]; Flatten[ Table[ row[n], {n, 1, 17}]] (* Jean-François Alcover, Jun 15 2012, after David Broadhurst *)

A259486 a(n) = 3n^2 - 3n + 1 + 6floor((n-1)(n-2)/6).

Original entry on oeis.org

1, 7, 19, 43, 73, 109, 157, 211, 271, 343, 421, 505, 601, 703, 811, 931, 1057, 1189, 1333, 1483, 1639, 1807, 1981, 2161, 2353, 2551, 2755, 2971, 3193, 3421, 3661, 3907, 4159, 4423, 4693, 4969, 5257, 5551, 5851, 6163, 6481, 6805, 7141, 7483, 7831, 8191, 8557

Offset: 1

Wesley Ivan Hurt, Jun 28 2015

Start with the geometric picture for the centered hex numbers (A003215). Here, each hexagonal figure in the sequence is the aggregate of smaller unit hexes (with n hexes along each side). Then, when possible, add additional unit hexes to each side except for the corners --> do this repeatedly with the same restriction until no hexes can be added. a(n) gives the area of each figure (see example).

a(n) == 1 mod 6. - Robert Israel, Jun 29 2015

			-----------------------------------------------------------------------
Figure 1
-----------------------------------------------------------------------
                                                  __    __    __
                                                 /  \__/  \__/  \
                                                 \_*/  \__/  \*_/
                              __               __/  \__/  \__/  \__
                           __/  \__           /  \__/  \__/  \__/  \
            __          __/  \__/  \__        \__/  \__/  \__/  \__/
         __/  \__      /  \__/  \__/  \     __/  \__/  \__/  \__/  \__
.__     /  \__/  \     \__/  \__/  \__/    / *\__/  \__/  \__/  \__/* \
/  \    \__/  \__/     /  \__/  \__/  \    \__/  \__/  \__/  \__/  \__/
\__/    /  \__/  \     \__/  \__/  \__/       \__/  \__/  \__/  \__/
        \__/  \__/     /  \__/  \__/  \       /  \__/  \__/  \__/  \
           \__/        \__/  \__/  \__/       \__/  \__/  \__/  \__/
                          \__/  \__/             \__/  \__/  \__/
                             \__/                / *\__/  \__/* \
                                                 \__/  \__/  \__/
n=1         n=2               n=3                       n=4
-----------------------------------------------------------------------
Table 1
-----------------------------------------------------------------------
a(1) = 1                              =  1
a(2) = 3  + 2(2)                      =  7
a(3) = 5  + 2(3+4)                    =  19
a(4) = 7  + 2(4+5+6)          + 6(1)  =  43
a(5) = 9  + 2(5+6+7+8)        + 6(2)  =  73
a(6) = 11 + 2(6+7+8+9+10)     + 6(3)  =  109
a(7) = 13 + 2(7+8+9+10+11+12) + 6(5)  =  157
...

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

Cf. A003215 (hex numbers), A000969, A130518, A255840 (similar, with squares).

[3*n^2 - 3*n + 1 + 6*Floor((n-1)*(n-2)/6) : n in [1..100]];

I:=[1,7,19,43,73]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

A259486:=n->3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6): seq(A259486(n), n=1..100);

Table[3 n^2 - 3 n + 1 + 6 Floor[(n - 1) (n - 2)/6], {n, 50}] (* or *)
CoefficientList[Series[(1 + 5 x + 6 x^2 + 11 x^3 + x^4)/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x]
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 7, 19, 43, 73}, 50]; (* Vincenzo Librandi, Jul 14 2015 *)

G.f.: (1+5*x+6*x^2+11*x^3+x^4)/((1-x)^3*(1+x+x^2)).
a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5), n>5.
a(n) = A003215(n+1) + 6*A130518(n+1).
From Robert Israel, Jun 29 2015: (Start)
a(n) = 4*n^2 - 6*n + 1 if 3 divides n, 4*n^2 - 6*n + 3 otherwise.
a(n) = 1 + 6 * A000969(n-2) for n >= 2. (End)
a(n) = 4*n^2 - 6*n + 3^sign(n mod 3). - Wesley Ivan Hurt, Jul 13 2015

A320259 Terms that are on the y-axis of the square spiral built with 2k, 2k+1, 2*k+1 for k >= 0.

Original entry on oeis.org

0, 2, 5, 9, 15, 22, 30, 40, 51, 63, 77, 92, 108, 126, 145, 165, 187, 210, 234, 260, 287, 315, 345, 376, 408, 442, 477, 513, 551, 590, 630, 672, 715, 759, 805, 852, 900, 950, 1001, 1053, 1107, 1162, 1218, 1276, 1335

Offset: 0

Paul Curtz, Oct 08 2018

nonn

a(n) mod 9 is of period 27.
The spiral:
     28--29--29--30--31--31--32
      |
     27  13--14--15--15--16--17
      |   |                   |
     27  13   4---5---5---6  17
      |   |   |           |   |
     26  12   3   0---1   7  18
      |   |   |       |   |   |
     25  11   3---2---1   7  19
      |   |               |   |
     25  11--10---9---9---8  19
      |                       |
     24--23--23--22--21--21--20

			G.f. = 2*x + 5*x^2 + 9*x^3 + 15*x^4 + 22*x^5 + 30*x^6 + ... - _Michael Somos_, Nov 13 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000969, A004396, A004523, A004767, A004772 (first differences), A211480, A002264, A143978.

a:=[0,2,5,9,15];; for n in [6..50] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-2*a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Oct 08 2018

seq(coeff(series(x^2*(2+x+x^2)/((1-x)^3*(1+x+x^2)),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 08 2018

LinearRecurrence[{2,-1,1,-2,1}, {0, 2, 5, 9, 15}, 50] (* or *)
CoefficientList[Series[x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 09 2018 *)
a[ n_] := Quotient[(n + 1) (2 n + 1), 3]; (* Michael Somos, Nov 13 2018 *)

concat(0, Vec(x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Oct 08 2018

{a(n) = (n + 1) * (2*n + 1) \ 3}; /* Michael Somos, Nov 13 2018 */

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), a(0)=0, a(1)=2, a(2)=5, a(3)=9, a(4)=15.
a(n) = a(n-1) + A004772(n+1), a(0)=0, n>0.
a(n+15) = a(n-15) + 10*A004767(n).
a(-n-1) = ({0} U A000969(n)) = 0, 1, 3, 7, ... = b(n), the full x-axis terms.
a(-n-1) + a(n) = 0, 3, 8, 16, ... = A211480(n+1).
a(n) = b(n) + A004523(n+1).
G.f.: x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Oct 08 2018
a(n) = A143978(n) + A002264(n+2).
a(n) = A000969(-2-n) for all n in Z. - Michael Somos, Nov 13 2018

A320281 Terms that are on the positive x-axis of the square spiral built with 2k, 2k+1, 2*k+1 for k >= 0.

Original entry on oeis.org

0, 1, 7, 18, 35, 57, 84, 117, 155, 198, 247, 301, 360, 425, 495, 570, 651, 737, 828, 925, 1027, 1134, 1247, 1365, 1488, 1617, 1751, 1890, 2035, 2185, 2340, 2501, 2667, 2838, 3015, 3197, 3384, 3577, 3775, 3978

Offset: 0

Paul Curtz, Oct 09 2018

Resulting spiral:
      28--29--29--30--31--31--32
       |
      27  13--14--15--15--16--17
       |   |                   |
      27  13   4---5---5---6  17
       |   |   |           |   |
      26  12   3   0---1   7  18
       |   |   |       |   |   |
      25  11   3---2---1   7  19
       |   |               |   |
      25  11--10---9---9---8  19
       |                       |
      24--23--23--22--21--21--20
.
a(n) mod 9 is of period 27. a(n) mod 10 is of period 30.
The NE diagonal starting at 1 is A301696. - Klaus Purath, May 15 2021

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000969.

LinearRecurrence[{2,-1,1,-2,1},{0,1,7,18,35},100] (* Paolo Xausa, Nov 13 2023 *)

concat(0, Vec(x*(1 + 5*x + 5*x^2 + 5*x^3) / ((1 - x)^3*(1 + x + x^2)) + O(x^50))) \\ Colin Barker, Oct 09 2018

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), a(0)=0, a(1)=1, a(2)=7, a(3)=18, a(4)=35.
a(n+2) - 2*a(n-1) + a(n) = period 3: repeat [5, 5, 6].
a(-n) = 0, 5, 15, 30, 51, 77, 108, 145, ... is the sequence of the terms on the positive y-axis.
G.f.: x*(1 + 5*x + 5*x^2 + 5*x^3) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Oct 09 2018

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A000970 Fermat coefficients.

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A000971 Fermat coefficients.

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A000972 Fermat coefficients.

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A011793 Triangle of numbers of irreducible Euler sums.

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A259486 a(n) = 3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6).

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A320259 Terms that are on the y-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.

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A320281 Terms that are on the positive x-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.

A259486 a(n) = 3n^2 - 3n + 1 + 6floor((n-1)(n-2)/6).

A320259 Terms that are on the y-axis of the square spiral built with 2k, 2k+1, 2*k+1 for k >= 0.

A320281 Terms that are on the positive x-axis of the square spiral built with 2k, 2k+1, 2*k+1 for k >= 0.