cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A155602 4^n + 3^n - 1.

Original entry on oeis.org

1, 6, 24, 90, 336, 1266, 4824, 18570, 72096, 281826, 1107624, 4371450, 17308656, 68703186, 273218424, 1088090730, 4338014016, 17309009346, 69106897224, 276040168410, 1102998412176, 4408506864306, 17623567104024, 70462887356490
Offset: 0

Views

Author

Mohammad K. Azarian, Jan 25 2009

Keywords

Links

Crossrefs

Cf. A074501, A020515, A155588, A155590, A155592-A155594, A155596-A155601.

Programs

Formula

G.f.: 1/(1-4*x)+1/(1-3*x)-1/(1-x). E.g.f.: e^(4*x)+e^(3*x)-e^x.
a(n) = 7*a(n-1) - 12*a(n-2) -6, n>1 - Gary Detlefs, Jun 21 2010
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3), n>2, a(0)=1, a(1)=6, a(2)=24. - L. Edson Jeffery, Oct 17 2012
a(n) = A074605(n)-1. - R. J. Mathar, Mar 10 2022

A074506 a(n) = 1^n + 3^n + 4^n.

Original entry on oeis.org

3, 8, 26, 92, 338, 1268, 4826, 18572, 72098, 281828, 1107626, 4371452, 17308658, 68703188, 273218426, 1088090732, 4338014018, 17309009348, 69106897226, 276040168412, 1102998412178, 4408506864308, 17623567104026
Offset: 0

Views

Author

Robert G. Wilson v, Aug 23 2002

Keywords

Links

Crossrefs

Cf. A001550, A001576, A034513, A001579, A074501..A074580.
Equals A074605(n) + 1.

Programs

  • Mathematica
    Table[1^n + 3^n + 4^n, {n, 0, 22}]
LinearRecurrence[{8,-19,12},{3,8,26},30] (* Harvey P. Dale, May 12 2025 *)

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 6 with a(0)=3, a(1)=8. - Vincenzo Librandi, Jul 19 2010
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - R. J. Mathar, Jul 18 2010
From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-3*x) + 1/(1-4*x).
E.g.f.: e^x + e^(3*x) + e^(4*x). (End)

A045584 Numbers k that divide 4^k + 3^k.

Original entry on oeis.org

1, 7, 49, 343, 2401, 2653, 16807, 18571, 117649, 129997, 823543, 909979, 1005487, 4941601, 5764801, 6369853, 7038409, 34591207, 40353607, 44588971, 49268863, 236474833, 242138449, 282475249, 312122797, 344882041, 381079573, 1655323831, 1694969143, 1872866779, 1977326743
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Cf. A074605.

Programs

  • Mathematica
    Select[Range[10^6], Divisible[PowerMod[3, #, #] + PowerMod[4, #, #], #] &] (* Amiram Eldar, Oct 23 2021 *)
  • PARI
    isok(k) = Mod(4, k)^k + Mod(3, k)^k == 0; \\ Michel Marcus, May 16 2022
  • Python
    from itertools import islice, count
def A045584_gen(startvalue=1): # generator of terms >= startvalue
    kstart = max(startvalue,1)
    k3, k4 = 3**kstart, 4**kstart
    for k in count(kstart):
        if (k3+k4) % k == 0:
            yield k
        k3 *= 3
        k4 *= 4
A045584_list = list(islice(A045584_gen(),10)) # Chai Wah Wu, May 16 2022

Extensions

a(28)-a(31) from Amiram Eldar, Oct 23 2021

A245806 a(n) = 3^n + 10^n.

Original entry on oeis.org

2, 13, 109, 1027, 10081, 100243, 1000729, 10002187, 100006561, 1000019683, 10000059049, 100000177147, 1000000531441, 10000001594323, 100000004782969, 1000000014348907, 10000000043046721, 100000000129140163, 1000000000387420489, 10000000001162261467
Offset: 0

Views

Author

Vincenzo Librandi, Aug 04 2014

Keywords

Links

Crossrefs

Cf. 3^n+k^n: A034472 (k=1), A007689 (k=2), A008776 (k=3), A074605 (k=4), A074606 (k=5), A074607 (k=6), A074608 (k=7), A074609 (k=8), A074610 (k=9), this sequence (k=10).

Programs

  • Magma
    [3^n+10^n: n in [0..25]];
  • Magma
    I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1)-30*Self(n-2): n in [1..25]];
  • Mathematica
    Table[(3^n + 10^n), {n, 0, 30}] (* or *) CoefficientList[Series[(2 - 13 x)/((1 - 3 x) (1 - 10 x)), {x, 0, 30}], x]
  • PARI
    a(n)=3^n + 10^n \\ Charles R Greathouse IV, Aug 26 2014

Formula

G.f.: (2-13*x)/((1-3*x)(1-10*x)).
E.g.f.: e^(3*x) + e^(10*x).
a(n) = 13*a(n-1)-30*a(n-2) for n>1.
a(n) = A000244(n) + A011557(n). - Michel Marcus, Aug 04 2014

A210694 T(n,k)=Number of (n+1)X(n+1) -k..k symmetric matrices with every 2X2 subblock having sum zero.

Original entry on oeis.org

5, 13, 9, 25, 35, 17, 41, 91, 97, 33, 61, 189, 337, 275, 65, 85, 341, 881, 1267, 793, 129, 113, 559, 1921, 4149, 4825, 2315, 257, 145, 855, 3697, 10901, 19721, 18571, 6817, 513, 181, 1241, 6497, 24583, 62281, 94509, 72097, 20195, 1025, 221, 1729, 10657, 49575
Offset: 1

Views

Author

R. H. Hardin, with R. J. Mathar in the Sequence Fans Mailing List, Mar 30 2012

Keywords

Comments

Table starts
...5....13.....25......41.......61.......85.......113.......145........181
...9....35.....91.....189......341......559.......855......1241.......1729
..17....97....337.....881.....1921.....3697......6497.....10657......16561
..33...275...1267....4149....10901....24583.....49575.....91817.....159049
..65...793...4825...19721....62281...164305....379793....793585....1531441
.129..2315..18571...94509...358061..1103479...2920695...6880121...14782969
.257..6817..72097..456161..2070241..7444417..22542017..59823937..143046721
.513.20195.281827.2215269.12030821.50431303.174571335.521638217.1387420489
Solutions are determined by the diagonal, extended with x(i,j) = (x(i,i)+x(j,j))/2 * (-1)^(i-j)

Examples

			Some solutions for n=3 k=4
.-2..1.-3..0....0.-1..0..1....4..0..1.-1....2.-1.-1.-2....3.-2..1..0
..1..0..2..1...-1..2.-1..0....0.-4..3.-3...-1..0..2..1...-2..1..0.-1
.-3..2.-4..1....0.-1..0..1....1..3.-2..2...-1..2.-4..1....1..0.-1..2
..0..1..1..2....1..0..1.-2...-1.-3..2.-2...-2..1..1..2....0.-1..2.-3

Links

Crossrefs

cf. A179927
Column 1 is A000051(n+1)
Column 2 is A007689(n+1)
Column 3 is A074605(n+1)
Column 4 is A074611(n+1)
Column 5 is A074615(n+1)
Column 6 is A074619(n+1)
Column 7 is A074622(n+1)
Column 8 is A074624(n+1)
Row 1 is A001844
Row 2 is A005898
Row 3 is A008514
Row 4 is A008515
Row 5 is A008516
Row 6 is A036085
Row 7 is A036086
Row 8 is A036087

Formula

T(n,k)=k^(n+1)+(k+1)^(n+1)

A213660 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the cycle graph C_3 with a vertex in common.

Original entry on oeis.org

3, 3, 1, 1, 8, 10, 5, 1, 1, 6, 23, 32, 21, 7, 1, 1, 8, 28, 72, 102, 80, 36, 9, 1, 1, 10, 45, 120, 242, 332, 290, 160, 55, 11, 1, 1, 12, 66, 220, 495, 856, 1116, 1032, 655, 280, 78, 13, 1, 1, 14, 91, 364, 1001, 2002, 3131, 3880, 3675, 2562, 1281, 448, 105, 15, 1
Offset: 1

Views

Author

Emeric Deutsch, Jun 29 2012

Keywords

Comments

Row n contain 2n + 1 entries.
Sum of entries in row n = 3^n + 4^n = A074605(n).

Examples

			Row 1 is 3,3,1 because the graph G(1) is the triangle abc; there are 3 dominating subsets of size 1 ({a}, {b}, {c}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
T(n,1)=1 for n >= 2 because the common vertex of the triangles is the only dominating subset of size k=1.
Triangle starts:
  3, 3,  1;
  1, 8, 10,  5,   1;
  1, 6, 23, 32,  21,  7,  1;
  1, 8, 28, 72, 102, 80, 36, 9, 1;

Links

Crossrefs

Cf. A074605.

Programs

  • Magma
    /* As triangle */ [[2^(2*n-k)*Binomial(n,k-n)+Binomial(2*n,k-1): k in [1..2*n+1]]: n in [1.. 10]]; // Vincenzo Librandi, Jul 20 2019
  • Maple
    T := proc (n, k) options operator, arrow: 2^(2*n-k)*binomial(n, k-n)+binomial(2*n, k-1) end proc: for n to 9 do seq(T(n, k), k = 1 .. 2*n+1) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := 2^(2n-k) Binomial[n, k-n] + Binomial[2n, k-1];
Table[T[n, k], {n, 1, 9}, {k, 1, 2n+1}] // Flatten (* Jean-François Alcover, Dec 06 2017 *)

Formula

Generating polynomial of row n is x*(1+x)^(2*n) + (2*x+x^2)^n; this is the domination polynomial of the graph G(n).
T(n,k) = 2^(2*n-k)*binomial(n,k-n) + binomial(2*n,k-1) (n >= 1; 1 <= k <= 2*n+1).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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