A111384 -id:A111384 - cp's OEIS frontend

A005997 Number of paraffins.

1, 3, 10, 20, 39, 63, 100, 144, 205, 275, 366, 468, 595, 735, 904, 1088, 1305, 1539, 1810, 2100, 2431, 2783, 3180, 3600, 4069, 4563, 5110, 5684, 6315, 6975, 7696, 8448, 9265, 10115, 11034, 11988, 13015, 14079, 15220, 16400, 17661, 18963, 20350, 21780, 23299

Offset: 1

N. J. A. Sloane

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
M. Golafshan, M. Rigo, and M. Whiteland, Computing the k-binomial complexity of generalized Thue-Morse words, arXiv:2412.18425 [math.CO], 2024. See p. 29.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A005999.

a:= n-> (Matrix([[0, 0, -1, -5, -12, -26]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=1..50); # Alois P. Heinz, Jul 31 2008

A005997[n_]:=1+Floor[(n-1)/2]+2*(Binomial[n+1,3]-Binomial[Floor[(n+1)/2],3]-Binomial[Ceiling[(n+1)/2],3]); Array[A005997,37] (* Enrique Pérez Herrero, Apr 22 2012 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 3, 10, 20, 39, 63}, 37] (* Bruno Berselli, Apr 22 2012 *)

G.f.: (x^3+3*x^2+x+1)*x / ((-1+x)^2*(-1+x^2)^2).
a(n) = A005999(n) + (n-1)^2. - Enrique Pérez Herrero, Mar 27 2012
a(n) = 1 + floor((n-1)/2) + 2*(C(n+1,3)-C(floor((n+1)/2),3)-C(ceiling((n+1)/2),3)). - Enrique Pérez Herrero, Apr 22 2012
a(n) = (n+1)*(2*n^2-(-1)^n+1)/8. - Bruno Berselli, Apr 22 2012
a(n) = A004526(n) + 2*A111384(n). - Enrique Pérez Herrero, Apr 25 2012
E.g.f.: (x*(3 + 4*x + x^2)*cosh(x) + (1 + 2*x + 4*x^2 + x^3)*sinh(x))/4. - Stefano Spezia, Dec 13 2021

A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 40, 70, 112, 176, 261, 372, 520, 704, 935, 1220, 1560, 1976, 2464, 3038, 3710, 4480, 5376, 6392, 7548, 8856, 10320, 11970, 13800, 15840, 18095, 20580, 23320, 26312, 29601, 33176, 37072, 41300, 45875, 50830, 56160, 61920, 68096, 74732

Offset: 1

Pontus von Brömssen, Mar 26 2022

nonn

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 1/2, the inducibility of the claw graph.

Brown and Sidorenko (1994) prove that a bipartite optimal graph (i.e., an n-node graph with a(n) induced claw graphs) exists for all n. For n >= 2, the size k of the smallest part of an optimal bipartite graph K_{k,n-k} is one of the two integers closest to n/2 - sqrt(3*n/4-1), and a(n) = binomial(k,3)*(n-k) + binomial(n-k,3)*k. Both are optimal if and only if n is in A271713. For 7 <= n <= 10 (and, trivially, n = 3), the tripartite graph K_{1,1,n-2} is also optimal.

Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.

Cf. A271713.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

from math import comb,isqrt
def A352666(n):
    if n <= 1: return 0
    r = isqrt(3*n-4)
    k0 = (n-r-1)//2
    return max(comb(k,3)*(n-k)+comb(n-k,3)*k for k in (k0,k0+1))

A352667 Maximum number of induced copies of the paw graph in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 18, 36, 60, 97, 152, 224

Offset: 1

Pontus von Brömssen, Mar 26 2022

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 3/8, the inducibility of the paw graph (Hirst 2014).
Assuming that the extremal graph is KK(j_1, j_2; k_1, k_2), as defined in the Example section below, for some j_1, j_2, k_1, k_2 (these graphs are asymptotically extremal), the sequence would continue as follows, with a(n) = (binomial(j_1,2)*j_2 + binomial(j_2,2)*j_1)*(k_1 + k_2) + (binomial(k_1,2)*k_2 + binomial(k_2,2)*k_1)*(j_1 + j_2).
   n |    a(n)     | (j_1, j_2), (k_1, k_2)
     |(conjectural)|
  -------------------------------------------------------
     316       (2,2), (4,5)
     440       (2,2), (5,5)
     590       (2,3), (5,5)
     780       (3,3), (5,5)
    1008       (2,3), (6,6), or (3,3), (5,6)
    1296       (3,3), (6,6)
    1620       (3,3), (6,7), or (3,4), (6,6)
    2016       (3,3), (7,7), or (4,4), (6,6)
    2478       (3,4), (7,7)
    3024       (4,4), (7,7)
    3632       (4,4), (7,8)
    4352       (4,4), (8,8)
    5152       (4,5), (8,8)
    6080       (5,5), (8,8)
    7100       (5,5), (8,9)
    8280       (5,5), (9,9)
    9558       (5,6), (9,9)
   11016       (6,6), (9,9)
   12600       (5,6), (10,10), or (6,6), (9,10)
   14400       (6,6), (10,10)
   16320       (6,6), (10,11), or (6,7), (10,10)
   18480       (6,6), (11,11), or (7,7), (10,10)
   20812       (6,7), (11,11)
   23408       (7,7), (11,11)
   26166       (7,7), (11,12)
   29232       (7,7), (12,12)
   32496       (7,8), (12,12)
   36096       (8,8), (12,12)
   39904       (8,8), (12,13)
   44096       (8,8), (13,13)
   48516       (8,9), (13,13)
   53352       (9,9), (13,13)
   58446       (9,9), (13,14)
   64008       (9,9), (14,14)
   69832       (9,10), (14,14)
   76160       (10,10), (14,14)
   82800       (9,10), (15,15), or (10,10), (14,15)
   90000       (10,10), (15,15)
For n > 10, more than one optimal graph of the form KK(j_1, j_2; k_1, k_2) seem to exist exactly when n = 2*m^2 + i, where m >= 3 and i = -1, 1, or 2.

			All extremal graphs (i.e., n-node graphs having a(n) induced paw graphs) for 4 <= n <= 12 are listed below. Here, KK(j_1, j_2; k_1, k_2) denotes the complement of the disjoint union of K_{j_1, j_2} and K_{k_1, k_2}.
  n = 4: KK(0,1;1,2) (the paw graph);
  n = 5: KK(0,1;2,2) (the butterfly graph);
  n = 6: KK(0,1;2,3);
  n = 7: KK(0,1;3,3), KK(0,2;2,3), and KK(1,1;2,3);
  n = 8: KK(0,2;3,3) and KK(1,1; 3,3);
  n = 9: KK(0,2;3,4), KK(1,1;3,4), and KK(1,2;3,3);
  n = 10: KK(1,2;3,4);
  n = 11: KK(1,2;4,4);
  n = 12: KK(2,2;4,4).

James Hirst, The inducibility of graphs on four vertices, Journal of Graph Theory 75 (2014), 231-243.
Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352668 (diamond graph), A352669 (cycles).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 05 2022

A352668 Maximum number of induced copies of the diamond graph K_{1,1,2} in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 24, 48, 84, 138, 216, 324, 459, 636, 864, 1152, 1488, 1900, 2400, 3000, 3675, 4470, 5400, 6480, 7668, 9030, 10584, 12348, 14259, 16408, 18816, 21504, 24384, 27576, 31104, 34992, 39123, 43650, 48600, 54000, 59700, 65890, 72600, 79860, 87483, 95742

Offset: 1

Pontus von Brömssen, Mar 26 2022

nonn

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 72/125, the inducibility of the diamond graph (Hirst 2014).

It is known that there exists a complete multipartite optimal graph (i.e., an n-node graph with a(n) induced diamond graphs) for all n (Brown and Sidorenko 1994) and that a complete 5-partite graph is asymptotically optimal (Hirst 2014). For 3 <= n <= 7, the 3-partite Turán graph is optimal, and for 7 <= n <= 45, the 4-partite Turán graph is optimal. (For n = 7 both are optimal.) It appears that the 5-partite Turán graph is optimal for all n >= 46 (verified up to n = 75).

Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.
James Hirst, The inducibility of graphs on four vertices, Journal of Graph Theory 75 (2014), 231-243.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352667 (paw graph), A352669 (cycles).

from sympy.utilities.iterables import combinations,partitions
from sympy.combinatorics import IntegerPartition
f=lambda p:sum(q[0]*q[1]*q[2]*(q[0]+q[1]+q[2]-3)//2 for q in combinations(p,3)) # number of induced diamond graphs in the multipartite graph with parts of sizes given by p
def A352668(n):
    return max(f(IntegerPartition(p).partition) for p in partitions(n))

A352669 Maximum number of induced cycles in an n-node graph.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 225

Offset: 1

Pontus von Brömssen, Mar 26 2022

For 3 <= n <= 11, a(n) = binomial(n,3) = A000292(n-2) and the complete graph is the unique extremal graph, but a(12) = 225 > binomial(12,3), where the unique extremal graph is K_{6,6}.

Morrison and Scott (2017) prove that, for sufficiently large n (they say it ought to be true for n >= 30), a(n) = A276401(n), with the unique extremal graph being the empty cyclic braid graph with one cluster of size 4 if n == 1 (mod 3), one cluster of size 2 if n == 2 (mod 3), and all other clusters of size 3. (The empty cyclic braid graph is obtained by arranging clusters of nodes of the appropriate sizes in a cycle and joining all pairs of nodes in neighboring clusters with edges.) For 14 <= n <= 21, this graph is not extremal, because the balanced bipartite graph K_{floor(n/2),ceiling(n/2)} has A028723(n+1) > A276401(n) induced cycles.

Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Cf. A000292, A028723, A276401.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352667 (paw graph), A352668 (diamond graph).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 07 2022

A352665 Maximum number of induced copies of the 4-node path in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 5, 9, 16, 32, 48, 80, 112, 160

Offset: 1

Pontus von Brömssen, Mar 26 2022

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to the inducibility of the 4-node path, which is known to be between 1173/5824 = 0.201407... and 0.204513; see Even-Zohar and Linial (2015), who attribute the upper bound to Emil R. Vaughan.

			All optimal graphs (i.e., n-node graphs having a(n) induced copies of P_4) for 4 <= n <= 9 are listed below. Since P_4 is self-complementary, the optimal graphs come in complementary pairs. Here, ECB(n_1, ..., n_k) denotes the empty cyclic braid graph with cluster sizes n_1, ..., n_k, as defined by Morrison and Scott (2017), i.e., the graph obtained by arranging k clusters of n_1, ..., n_k nodes, respectively, in a cycle, and joining all pairs of nodes in neighboring clusters with edges.
  n = 4: P_4 (self-complementary).
  n = 5: C_5 (self-complementary).
  n = 6: ECB(1, 1, 1, 1, 2) and its complement.
  n = 7: 8 optimal graphs, among them ECB(1, 1, 1, 2, 2) and ECB(1, 1, 2, 1, 2), and their complements. In graph6 format, the optimal graphs are "F?o~_", "FCY^_", "FCpv?", "FCxv?", "FCxvO", "FQjRo", "FQyuo", and "FQyvO".
  n = 8: The antiprism graph and its complement (the Wagner graph).
  n = 9: 22 optimal graphs, among them all graphs that are supergraphs of ECB(1, 2, 2, 2, 2) and subgraphs of its complement (10 graphs altogether), and the 1-skeletons of the Johnson solids J10 (the gyroelongated square pyramid) and J51 (the triaugmented triangular prism) and their complements. In graph6 format, the optimal graphs are "H?bF`xw", "H?o}^_}", "H?o}^bp", "H?q`qjo", "H?q`v`[", "H?rF`zo", "H?rF`zq", "HCRbdO{", "HCXfczo", "HCXfczq", "HCXk~a]", "HCXk~bo", "HCXk~bp", "HCY^fXy", "HCrb`qi", "HCrb`rc", "HEhuTxm", "HEhutxm", "HQjUjqm", "HQyurjU", "HQyurji", and "HQyurzU".

Chaim Even-Zohar and Nati Linial, A Note on the inducibility of 4-vertex graphs, Graphs and Combinatorics 31 (2015), 1367-1380; arXiv version, arXiv:1312.1205 [math.CO], 2013-2014.
Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352666 (claw graph), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 07 2022

A120714 Expansion of 2x^2(7+14x+4x^2)/((1+x-x^2)(1-2x-10x^2-6x^3)).

Original entry on oeis.org

0, 14, 42, 232, 974, 4522, 20180, 91422, 411782, 1858856, 8384078, 37827386, 170648724, 769875718, 3473203086, 15669055544, 70689396502, 318908566562, 1438725432052, 6490672907694, 29282051536966, 132103184740456

Offset: 1

Roger L. Bagula, Aug 12 2006

Previous name was: Sequence produced by 7 X 7 Markov chain based on adjacency matrix of 7-vertex graph with 10 edges, derived from the Fano plane.
Take the standard 7-vertex 7-edge Fano plane graph and add three edges that go around the triangle vertices from the middle of the sides ( connecting the middle of the sides without going through the center)
Characteristic polynomial is 6 - 2*x - 24*x^2 - 3*x^3 + 26*x^4 + 15*x^5 - x^7.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fano Plane
Index entries for linear recurrences with constant coefficients, signature (0,15,26,-3,-24,-2,6).

Cf. A111384, A120715.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)) )); // G. C. Greubel, Jul 22 2023

a[1]:=0: a[2]:=14: a[3]:=42: a[4]:=232: a[5]:=974: a[6]:=4522: a[7]:=20180: a[8]:=91422: for n from 9 to 25 do a[n]:=15*a[n-2]+26*a[n-3]-3*a[n-4]-24*a[n-5]-2*a[n-6]+6*a[n-7] end do: seq(a[n], n=1..25);

LinearRecurrence[{0,15,26,-3,-24,-2,6},{0,14,42,232,974,4522,20180},30] (* Harvey P. Dale, Sep 20 2011 *)

def A120714_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)) ).list()
a=A120714_list(30); a[1:] # G. C. Greubel, Jul 22 2023

a(n) = 15*a(n-2) +26*a(n-3) -3*a(n-4) -24*a(n-5) -2*a(n-6) +6*a(n-7).

O.g.f.: 2*x^2*(7+14*x+4*x^2)/((1+x-x^2)*(1-2*x-10*x^2-6*x^3)). - R. J. Mathar, Dec 05 2007

Edited by N. J. A. Sloane, Jul 14 2007, Jul 28 2007

New name using g.f. from Joerg Arndt, Sep 21 2021

A120715 Sequence produced by 14 X 14 Markov chain based on 14-vertex graph formed from direct product of two copies of the graph used in A120714.

Original entry on oeis.org

0, 27, 838, 4025, 29742, 161630, 962784, 5335471, 30120946, 166834881, 926998480, 5122817760, 28316610392, 156260679433, 862162027134, 4754345230927, 26214240435218, 144511100239056, 796592187757696

Offset: 0

Roger L. Bagula, Aug 12 2006, corrected Jul 14 2007

Characteristic polynomial: -17 - 96*x + 65*x^2 + 1528*x^3 + 3840*x^4 + 2996*x^5 - 1566*x^6 - 3312*x^7 - 702*x^8 + 880*x^9 + 372*x^10 - 52*x^11 - 37*x^12 + x^14.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Fano Plane
Index entries for linear recurrences with constant coefficients, signature (2,30,-6,-263,-250,419,666,228,-28,-17).

Cf. A111384.

R:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x*(27+784*x+1539*x^2-3286*x^3-6475*x^4-1442*x^5-3783*x^6-4444*x^7 -986*x^8)/((1-x-x^2)*(1+3*x+x^2)*(1-5*x-3*x^2+x^3)*(1+x-11*x^2 -17*x^3)) )); // G. C. Greubel, Jul 22 2023

M = {{0,1,0,0,0,1,1,1,0,0,0,0,0,0}, {1,0,1,1,0,1,1,0,1,0,0,0,0,0}, {0, 1,0,1,0,0,1,0,0,1,0,0,0,0}, {0,1,1,0,1,1,1,0,0,0,1,0,0,0}, {0,0,0,1, 0,1,1,0,0,0,0,1,0,0}, {1,1,0,1,1,0,1,0,0,0,0,0,1,0}, {1,1,1,1,1,1,0, 0,0,0,0,0,0,1}, {1,0,0,0,0,0,0,0,1,0,0,0,1,1}, {0,1,0,0,0,0,0,1,0,1, 1,0,1,1}, {0,0,1,0,0,0,0,0,1,0,1,0,0,1}, {0,0,0,1,0,0,0,0,1,1,0,1,1, 1}, {0,0,0,0,1,0,0,0,0,0,1,0,1,1}, {0,0,0,0,0,1,0,1,1,0,1,1,0,1}, {0, 0,0,0,0,0,1,1,1,1,1,1,1,0}};
v[1]= Table[Fibonacci[n], {n,0,13}]; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n,50}]
LinearRecurrence[{2,30,-6,-263,-250,419,666,228,-28,-17}, {0,27,838, 4025,29742,161630,962784,5335471,30120946,166834881}, 50] (* G. C. Greubel, Jul 22 2023 *)

def f(x): return x*(27+784*x+1539*x^2-3286*x^3-6475*x^4-1442*x^5-3783*x^6-4444*x^7 -986*x^8)/((1-x-x^2)*(1+3*x+x^2)*(1-5*x-3*x^2+x^3)*(1+x-11*x^2 -17*x^3))
def A120715_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A120715_list(50) # G. C. Greubel, Jul 22 2023

G.f.: x*(27 +784*x +1539*x^2 -3286*x^3 -6475*x^4 -1442*x^5 -3783*x^6 -4444*x^7 -986*x^8)/((1 -x -x^2)*(1 +3*x +x^2)*(1 -5*x -3*x^2 +x^3)*(1 +x -11*x^2 -17*x^3)). - Colin Barker, Nov 29 2012

Edited by N. J. A. Sloane, Jul 14 2007

A385403 Minimum number of triples that cover {1..n}, such that every 2-coloring of {1..n} results in at least one monochromatic triple.

Original entry on oeis.org

10, 10, 7, 8, 8, 8

Offset: 5

David Dewan, Jun 27 2025

There is no solution for n<=4. For example for n=4, the coloring {1=red, 2=red, 3=blue, 4=blue} prevents any monochromatic triple.

This sequence gives the minimum number of triples covering {1..n} that force a monochromatic triple under any 2-coloring. For the contrasting maximum number of triples covering {1..n} that can avoid monochromatic triples under some 2-coloring, see A111384.

			a(6)=10. For example, these ten triples {{1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,6}, {1,5,6}, {2,3,6}, {2,4,5}, {3,4,5}} cover {1..6} and have at least one monochromatic triple for each of the 2^6 = 64 different 2-colorings of {1..6}.
The minimum is 10 because for each of the 167,960 different groups of 9 triples from the C(6,3) = 20 possible triples, there exists a 2-coloring of {1..6} that results in no monochromatic triple.
a(7)=7. For example, these seven triples {{1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6}} cover {1..7} and have at least one monochromatic triple for each of the 2^7 = 128 different 2-colorings of {1..7}.
The minimum is 7 because for each of the 1,623,160 different groups of 6 triples from the C(7,3) = 35 possible triples, there exists a 2-coloring of {1..7} that results in no monochromatic triple.

David Dewan, Mathematica program

Cf. A111384, A383181.

Mathematica
```
(* see Links *)
```

For n>=7, a(n) <= 7+ceiling((n-7)/3) by extending any minimal 7-triple solution on {1..7} with ceiling((n-7)/3) additional triples covering {8..n}. For example, a(10) <= 8 with {{1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6}, {8,9,10}}.

a(10) from Jinyuan Wang, Jun 28 2025

A120711 Expansion of 2x(7+16x-2x^2-14x^3)/(1-11x^2-12x^3+10x^4+12*x^5).

Original entry on oeis.org

0, 14, 32, 150, 492, 1894, 6724, 24854, 89972, 329238, 1197972, 4372054, 15930580, 58096214, 211770452, 772129110, 2814859092, 10262536534, 37414140244, 136403674454, 497291840852, 1813006427478, 6609762501972, 24097566365014

Offset: 0

Roger L. Bagula, Aug 12 2006

Former title: 7 X 7 matrix Matrov of seven vertex Fano Plane: Characteristic polynomial: 12 + 10*x - 24*x^2 - 21*x^3 + 12*x^4 + 12*x^5 - x^7.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Fano Plane
Index entries for linear recurrences with constant coefficients, signature (0,11,12,-10,-12).

Cf. A083099, A111384.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 2*x*(7+16*x-2*x^2-14*x^3)/(1-11*x^2-12*x^3+10*x^4+12*x^5) )); // G. C. Greubel, Jul 22 2023

M = {{0,1,0,0,0,1,1}, {1,0,1,0,0,0,1}, {0,1,0,1,0,0,1}, {0,0,1,0,1,0, 1}, {0,0,0,1,0,1,1}, {1,0,0,0,1,0,1}, {1,1,1,1,1,1,0}};
v[1] = {0,1,1,2,3,5,8}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n,50}]
LinearRecurrence[{0,11,12,-10,-12}, {0,14,32,150,492}, 40] (* G. C. Greubel, Jul 22 2023 *)

A083099=BinaryRecurrenceSequence(2,6,0,1)
def A120711(n): return (1/3)*(-1 -3*(-1)^n +(-2)^(n+1) +6*(A083099(n+1) +4*A083099(n)))
[A120711(n) for n in range(41)] # G. C. Greubel, Jul 22 2023

a(n) = 11*a(n-2) + 12*a(n-3) - 10*a(n-4) - 12*a(n-5).
G.f.: 2*x*(7+16*x-2*x^2-14*x^3)/((1-x)*(1+x)*(1+2*x)*(1-2*x-6*x^2)). - Colin Barker, Mar 26 2012
a(n) = (1/3)*(-1 - 3*(-1)^n + (-2)^(n+1) + 6*(A083099(n+1) + 4*A083099(n))). - G. C. Greubel, Jul 22 2023

Edited by G. C. Greubel, Jul 22 2023

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