A054490 -id:A054490 - cp's OEIS frontend

A154348 a(n) = 16a(n-1) - 56a(n-2) for n>1, with a(0)=1, a(1)=16.

1, 16, 200, 2304, 25664, 281600, 3068416, 33325056, 361369600, 3915710464, 42414669824, 459354931200, 4974457389056, 53867442077696, 583309459456000, 6316374594945024, 68396663789584384, 740629643316428800

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

Third binomial transform of A164609, fourth binomial transform of A164608, fifth binomial transform of A054490, sixth binomial transform of A164607, seventh binomial transform of A083100, eighth binomial transform of A164683.

lim_{n -> infinity} a(n)/a(n-1) = 8 + 2*sqrt(2) = 10.8284271247....

R. J. Mathar, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-56).

Cf. A002193 (decimal expansion of sqrt(2)), A164609, A164608, A054490, A164607, A083100, A164683.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+2*r)^n-(8-2*r)^n)/(4*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009

Join[{a=1,b=16},Table[c=16*b-56*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
LinearRecurrence[{16,-56},{1,16},30] (* Harvey P. Dale, Aug 31 2016 *)

a(n) = 16*a(n-1) - 56*a(n-2) for n>1. - Philippe Deléham, Jan 12 2009
a(n) = ( (8 + 2*sqrt(2))^n - (8 - 2*sqrt(2))^n )/(4*sqrt(2)).
G.f.: 1/(1 - 16*x + 56*x^2). - Klaus Brockhaus, Jan 12 2009; corrected Oct 08 2009
E.g.f.: (1/(2*sqrt(2)))*exp(8*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009
Edited by Klaus Brockhaus, Oct 08 2009
Offset corrected. - R. J. Mathar, Jun 19 2021

A210695 a(n) = 6*a(n-1) - a(n-2) + 6 with n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 12, 77, 456, 2665, 15540, 90581, 527952, 3077137, 17934876, 104532125, 609257880, 3551015161, 20696833092, 120629983397, 703083067296, 4097868420385, 23884127455020, 139206896309741, 811357250403432, 4728936606110857, 27562262386261716

Offset: 0

Kenneth J Ramsey, May 09 2012

It appears that if p is a prime of the form 8*r +/- 1, a(p-1) == 0 (mod p); and that if p is a prime of the form 8*r +/- 3, a(p+1) == 0 (mod p).

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A054490 (first differences).

[n le 2 select n-1 else 6*Self(n-1)-Self(n-2)+6: n in [1..23]]; // Bruno Berselli, Jun 26 2012

m = 36;n = 5; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m + 6; Sow[t];m = n; n = t;c++]][[2,1]]

G.f.: x*(1+5*x)/((1-x)*(1-6*x+x^2)). [Bruno Berselli, Jun 26 2012]
a(n) = ((1-2*sqrt(2))*(1-sqrt(2))^(2n-1)+(1+2*sqrt(2))*(1+sqrt(2))^(2n-1)-6)/4. [Bruno Berselli, Jun 26 2012]
2*a(n) = 3*A001109(n+1)-13*A001109(n)-3. - R. J. Mathar, Jul 18 2012

Definition rewritten from Bruno Berselli, Jun 26 2012

A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.

Original entry on oeis.org

1, 15, 117, 891, 6777, 51543, 392013, 2981475, 22675761, 172461663, 1311666021, 9975943179, 75872547369, 577052549415, 4388802753213, 33379264377459, 253867706760033, 1930803860947887, 14684827767302997, 111686210555580315, 849435201142733529, 6460422977475127287

Offset: 0

George Strand Vajagich, Mar 01 2020

			For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 2 to get 102 add 15 to get 117.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 2 to get 774 add 177 to get 891.

Colin Barker, Table of n, a(n) for n = 0..1000
Thezebraherd user, Graph W multiplication, Youtube video.
Index entries for linear recurrences with constant coefficients, signature (8,-3).

Cf. A332936 (number of blue nodes).

Similar sequences with a cycle size 3..6 are: A007483, A048876, A189274(n+1), A054490.

Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ Colin Barker, Mar 03 2020

g=1
b=7
sg=0
sb=0
bl=[]
gl=[]
for int in range(1,20):
  sg=g*1+b*2
  sb=b*7+g*2
  g=sg
  b=sb
  gl.append(g)
  bl.append(b)
print(gl)

a(n) = a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) + 7*b(n-1) with a(0) = 1 and b(0) = 7 where b(n) = A332936(n).
From Colin Barker, Mar 03 2020: (Start)
G.f.: (1 + 7*x) / (1 - 8*x + 3*x^2).
a(n) = 8*a(n-1) - 3*a(n-2) for n>1.
(End)
From Stefano Spezia, Mar 03 2020: (Start)
a(n) = ((4 - sqrt(13))^n*(-11 + sqrt(13)) + (4 + sqrt(13))^n*(11 + sqrt(13)))/(2*sqrt(13)).
E.g.f.: exp(4*x)*cosh(sqrt(13)*x) + (11*exp(4*x)*sinh(sqrt(13)*x))/sqrt(13).
(End)

a(14)-a(21) from Stefano Spezia, Mar 03 2020

Typo in a(14) fixed by Colin Barker, Apr 26 2020

A334456 Number h of points and of blocks of nontrivial biplanes.

Original entry on oeis.org

7, 11, 16, 37, 56, 79, 121

Offset: 1

Stefano Spezia, Apr 30 2020

A biplane is an incidence structure consisting of a set P of h points and a set of h blocks, each of which is a k-subset of P (block size k), such that every pair of points lies in exactly 2 blocks (see Alavi et al.).

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See Introduction p. 1.
P. Kaski and P. R. J. Östergård. There are exactly five biplanes with k = 11, J. Combin. Des., 16(2):117-127, 2008.

Cf. A001109, A003409, A038723, A038725, A054490, A334457 (k).

A334457 Block sizes k of nontrivial biplanes.

Original entry on oeis.org

4, 5, 6, 9, 11, 13, 16

Offset: 1

Stefano Spezia, Apr 30 2020

A biplane is an incidence structure consisting of a set P of h points and a set of h blocks, each of which is an k-subset of P (block size k), such that every pair of points lies in exactly 2 blocks (see Alavi et al.).

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See Introduction p. 1.
P. Kaski and P. R. J. Östergård. There are exactly five biplanes with k = 11, J. Combin. Des., 16(2):117-127, 2008.

Cf. A001109, A003409, A038723, A038725, A054490, A334456 (h).

cp's OEIS Frontend

A154348 a(n) = 16*a(n-1) - 56*a(n-2) for n>1, with a(0)=1, a(1)=16.

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A210695 a(n) = 6*a(n-1) - a(n-2) + 6 with n>1, a(0)=0, a(1)=1.

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A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.

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A334456 Number h of points and of blocks of nontrivial biplanes.

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A334457 Block sizes k of nontrivial biplanes.

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A154348 a(n) = 16a(n-1) - 56a(n-2) for n>1, with a(0)=1, a(1)=16.