A053698 -id:A053698 - cp's OEIS frontend

A348461 Size of largest bipartite biregular Moore graph of diameter 4 and degrees n and n.

8, 30, 80, 170, 312

Offset: 2

N. J. A. Sloane, Oct 31 2021

a(7) >= 516, a(8) = 800, a(9) = 1170, a(10) = 1640.

G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, arXiv:2103.11443 [math.CO], 2021.
G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, Discrete Math., 334 (2021), # 112582.

Empirical observation: For the terms a(2)-a(6) and a(8)-a(10)  a(n) = 2*(A027444(n-1) + 1).  It is unknown whether this is also valid for n = 7 and n > 10. - Hugo Pfoertner, Oct 31 2021
Is this the same as 2*A053698(n-1)? If not, where is the first place these sequences differ? - Omar E. Pol, Oct 31 2021
a(n) <= 2*A053698(n-1) (the Moore bound). - Pontus von Brömssen, Oct 31 2021

A066792 a(n) = phi(n^3 + n^2 + n + 1).

Original entry on oeis.org

1, 2, 8, 16, 64, 48, 216, 160, 288, 320, 1000, 480, 1344, 768, 1568, 1792, 4096, 1344, 4320, 2880, 4800, 3840, 8448, 3328, 11520, 7488, 12168, 6912, 17472, 6720, 24960, 13824, 16000, 13824, 25344, 14688, 46656, 19584, 26112, 24320, 64000, 19488

Offset: 0

Benoit Cloitre, Jan 18 2002

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000010 (phi), A053698, A243379.

Prepend[EulerPhi[Total[#^Range[0,3]]]&/@Range[45],1]  (* Harvey P. Dale, Feb 19 2011 *)

a(n) = eulerphi(n^3 + n^2 + n + 1); \\ Harry J. Smith, Mar 27 2010

a(n) = A000010(A053698(n)). - Michel Marcus, Sep 06 2022

Sum_{k=1..n} a(k) = c * n^4 + O((n*log(n))^3), where c = (3/16) * Product_{primes p == 1 (mod 4)} (1 - 3/p^2) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2) = 0.13549316168... . - Amiram Eldar, Dec 09 2024

A095797 Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

Original entry on oeis.org

1, 1, 1, 1, 4, 11, 14, 6, 35, 75, 70, 24, 204, 540, 570, 210, 1524, 3618, 3528, 1224, 9894, 25050, 25524, 9144, 69612, 169932, 168828, 59364, 467736, 1165908, 1175208, 417672, 3226524, 7947084, 7944648, 2806416, 21924672, 54371568, 54612456, 19359144, 150267840, 371199864

Offset: 0

Gary W. Adamson, Jun 06 2004

(n+1)-st set of 4 terms = leftmost finite differences of sequences generated from 3rd degree polynomials having n-th row coefficients, (given n = 1,2,3...) For example, first row is (1 1 1 1) with a corresponding polynomial x^3 + x^2 + x + 1. (f(x),x = 1,2,3...) = 4, 15, 40, 85, 156...Leftmost term of the sequence = 4, with finite difference rows: 11, 25, 45, 71...; 14, 20, 26, 32...; and 6, 6, 6, 6. Thus leftmost terms of the sequence 4, 15, 40...and the finite difference rows are (4 11 14 6) which is the second row.

The matrix generator is discussed in A028246, while 2nd degree polynomial examples are A091140, A091141 and A091140. The first degree case is A095795.

			3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].
Array begins:
     1,    1,    1,   1;
     4,   11,   14,   6;
    35,   75,   70,  24;
   204,  540,  570, 210;
  1524, 3618, 3528,1224;
  9894,25050,25524,9144;

Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).

Cf. A028246, A091140, A091141, A091142, A095795, A053698.

M := Matrix(4,4,[1,1,1,1,7,3,1,0,12,2,0,0,6,0,0,0]) ;
v := Vector(4,[1,1,1,1]) ;
for i from 0 to 20 do
        Mpr := (M ^ i).v ;
        for j from 1 to 4 do
                printf("%d,", Mpr[j]) ;
        end do;
end do; # R. J. Mathar, Jun 20 2011

LinearRecurrence[{0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12},{1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210},50] (* Harvey P. Dale, Feb 08 2013 *)

Vec((1+x+x^2+x^3+7*x^5+10*x^6+2*x^7-5*x^8+7*x^9-10*x^10-2*x^12 +6*x^13-16*x^14-24*x^11) / (1-4*x^4-24*x^8+30*x^12+12*x^16)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

G.f.: ( 1 +x +x^2 +x^3 +7*x^5 +10*x^6 +2*x^7 -5*x^8 +7*x^9 -10*x^10 -2*x^12 +6*x^13 -16*x^14 -24*x^11 ) / ( 1-4*x^4-24*x^8+30*x^12+12*x^16 ). - R. J. Mathar, Jun 20 2011

a(n) = +4*a(n-4) +24*a(n-8) -30*a(n-12) -12*a(n-16).

Name added by R. J. Mathar, several entries corrected by Charles R Greathouse IV, Jun 21 2011

A129801 Triangle read by rows in which row m (m>=0) gives the numbers 2mn + 1 for n = 0, ..., m.

Original entry on oeis.org

1, 1, 3, 1, 5, 9, 1, 7, 13, 19, 1, 9, 17, 25, 33, 1, 11, 21, 31, 41, 51, 1, 13, 25, 37, 49, 61, 73, 1, 15, 29, 43, 57, 71, 85, 99, 1, 17, 33, 49, 65, 81, 97, 113, 129, 1, 19, 37, 55, 73, 91, 109, 127, 145, 163, 1, 21, 41, 61, 81, 101, 121, 141, 161, 181, 201

Offset: 1

Roger L. Bagula, May 18 2007

Row sums are given by A053698 = n^3 + n^2 + n + 1.

			1; 1,3; 1,5,9; 1,7,13,19; 1,9,17,25,33; ...

Cf. A053698.

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 20 2007

A176071 Numbers of the form 2^k + k + 1 that are the product of two distinct primes.

Original entry on oeis.org

21, 38, 265, 4109, 65553, 262163, 1048597, 67108891, 274877906983, 4503599627370549, 73786976294838206531, 75557863725914323419213, 302231454903657293676623, 5192296858534827628530496329220209, 10889035741470030830827987437816582766726, 95780971304118053647396689196894323976171195136475313

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 07 2010

nonn

			21 = 3 * 7 = 2^4 + 4 + 1

Cf. A005126, A053698, A061421, A174969, A176069, A176070.

f[n_]:=Last/@FactorInteger[n]=={1,1};Select[Array[2^#+#+1&,140,0],f[ # ]&]
Select[Table[2^k+k+1,{k,0,200}],PrimeNu[#]==PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 11 2023 *)

is(n) = my(f = factor(n), e = logint(n, 2)); f[,2] == [1, 1]~ && n == 1<David A. Corneth, May 27 2023

Name corrected by David A. Corneth, May 27 2023

cp's OEIS Frontend

A348461 Size of largest bipartite biregular Moore graph of diameter 4 and degrees n and n.

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A066792 a(n) = phi(n^3 + n^2 + n + 1).

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A095797 Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

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A129801 Triangle read by rows in which row m (m>=0) gives the numbers 2*m*n + 1 for n = 0, ..., m.

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Extensions

A176071 Numbers of the form 2^k + k + 1 that are the product of two distinct primes.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A129801 Triangle read by rows in which row m (m>=0) gives the numbers 2mn + 1 for n = 0, ..., m.