A117808 -id:A117808 - cp's OEIS frontend

A108946 a(2n) = A001570(n), a(2n+1) = -A007654(n+1).

1, -3, 13, -48, 181, -675, 2521, -9408, 35113, -131043, 489061, -1825200, 6811741, -25421763, 94875313, -354079488, 1321442641, -4931691075, 18405321661, -68689595568, 256353060613, -956722646883, 3570537526921, -13325427460800, 49731172316281

Offset: 0

Creighton Dement, Jul 21 2005

In reference to program code, 2baseiseq[X](n) = ((-1)^n)*A001353(n) (a(n)^2 + 1 is a perfect square.) 1tesseq[X](n) = (-1^(n+1))*A097948(n).

Floretion Algebra Multiplication Program, FAMP Code: 1ibaseiseq[X] with X = .5'i + .5i' + 'ii' - .5'jj' + 1.5'kk' - 1 (* Corrected by Creighton Dement, Dec 11 2009 *)

Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (-4,0,4,1).

Cf. A007654, A001570, A076139. See also A117808, A122571 (same except for signs).

Cf. A001353, A097948.

/* By definition: */
m:=15; R:=PowerSeriesRing(Integers(), m);
A001570:=Coefficients(R!((1-x)/(1-14*x+x^2)));
A007654:=Coefficients(R!(-3*x^2*(1+x)/(-1+x)/(1-14*x+x^2)));
&cat[[A001570[i],-A007654[i]]: i in [1..m-2]]; // Bruno Berselli, Feb 05 2013

seriestolist(series((x^2+x+1)/((1-x)*(x+1)*(x^2+4*x+1)), x=0,25));

LinearRecurrence[{-4,0,4,1},{1,-3,13,-48},30] (* Harvey P. Dale, Jun 15 2018 *)

G.f.: (x^2+x+1)/((1-x)*(x+1)*(x^2+4*x+1)).
Floor(((2 + sqrt(3))^n + (2 - sqrt(3))^n)/4) produces this sequence with a different offset and without signs. - James R. Buddenhagen, May 20 2010
Define c(n) = a(n) - 4*a(n+1) - a(n+2) and d(n) = -a(n) - 4*a(n+1) - a(n+2); Conjectures: I: c(2n) = 24*A076139(n); (Triangular numbers that are one-third of another triangular number) II: c(2n+1) = -A011943(n+1); (Numbers n such that any group of n consecutive integers has integral standard deviation) III: d(2n) = -2; IV: d(2n+1) = -1

A239902 Exceptional primes: those for which Eq. (4.8) in Cosgrave and Dilcher (2011) fails.

Original entry on oeis.org

13, 181, 2521, 76543, 489061, 6811741, 1321442641, 18405321661, 381765135195632792959100810331957408101589361

Offset: 1

N. J. A. Sloane, Apr 06 2014

Comments from Christopher M. Stokes, Aug 02 2022: (Start)
Also the primes p for which A047788(p-1) = 0 mod p^2.
Also the primes for which the cyclotomic lambda invariant of Q(sqrt{-3}) is greater than 1. (End)

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.6.

J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.
John B. Cosgrave and Karl Dilcher, The multiplicative orders of certain Gauss factorials II, Funct. Approx. Comment. Math. Volume 54, Number 1 (2016), 73-93.
D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, Computation of Iwasawa lambda invariants for imaginary quadratic fields, Journal of number theory, 37(1) (1991), 100-121. [Reference added by _N. J. A. Sloane_, Jun 24 2022]
Christopher Stokes, On Gauss factorials and their application to Iwasawa theory for imaginary quadratic fields, arXiv:2207.07804 [math.NT], 2022.

Cf. A354772, A117808.

More terms from Cosgrave (2022), Section 18.6 added by N. J. A. Sloane, May 29 2022

a(9) from Stokes (2022) added by Michel Marcus, Jul 20 2022

A198196 Numbers k such that [V(4,1,k)/4] = floor(cosh(k log(sqrt(3)+2))/2) = A004526(A001075(k)) is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 79, 151, 199, 233, 251, 317, 863, 971

Offset: 1

M. F. Hasler, Oct 21 2011

Further terms are 3049, 7451, 7487, 18869.

All terms must be primes. A117808 lists the corresponding primes floor(A001075(k)/2). Actually, the "floor" here and in the definition is only needed for the initial term 2, since no other even (thus composite) k can be in the sequence, and A001075(k) is even for odd k.

D. Broadhurst, Re: Cubic factors of n^n+(n+1)^(n+1), in "primenumbers" Yahoo group, Oct 21 2011.
David Broadhurst, Cubic factors of n^n+(n+1)^(n+1), digest of 3 messages in primenumbers Yahoo group, Oct 21 - Oct 22, 2011.

q=2+quadgen(12); for(n=1,1e9, ispseudoprime(real(q^n)\2) & print1(n","))

cp's OEIS Frontend

A108946 a(2n) = A001570(n), a(2n+1) = -A007654(n+1).

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A239902 Exceptional primes: those for which Eq. (4.8) in Cosgrave and Dilcher (2011) fails.

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A198196 Numbers k such that [V(4,1,k)/4] = floor(cosh(k log(sqrt(3)+2))/2) = A004526(A001075(k)) is a prime.

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