A038723 -id:A038723 - cp's OEIS frontend

A111648 a(n) = A001541(n)^2 + A001653(n+1)^2 + A002315(n)^2.

3, 83, 2811, 95483, 3243603, 110187011, 3743114763, 127155714923, 4319551192611, 146737584833843, 4984758333158043, 169335045742539611, 5752406796913188723, 195412496049305876963

Offset: 0

Charlie Marion, Aug 24 2005

			a(1) = 83 = 3^2+5^2+7^2.

Ray Chandler, Table of n, a(n) for n = 0..652
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001541, A001653, A002315, A111647, A111649.

LinearRecurrence[{35, -35, 1}, {3, 83, 2811}, 20] (* Paolo Xausa, Feb 06 2024 *)

a(n) = A038761(n)^2 + 2, e.g., 95483 = 309^2 + 2.
a(n) = A001652(2*n+1) - A001109(n+1)^2 - Sum_{k=1..n-1} A038723(2*n), e.g., 95483 = 137903 - 204^2 - (23 + 781).
For n > 0, 2*a(n) + A001652(2*n-1) = A001653(2*n+2), e.g., 2*2811 + 119 = 5741.
G.f.: -(11*x^2-22*x+3) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Dec 14 2014 (Empirical g.f. confirmed for more terms and recurrence of source sequences. - Ray Chandler, Feb 05 2024)

A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.

Original entry on oeis.org

-1, 5, 43, 265, 1559, 9101, 53059, 309265, 1802543, 10506005, 61233499, 356895001, 2080136519, 12123924125, 70663408243, 411856525345, 2400475743839, 13990997937701, 81545511882379, 475282073356585, 2770146928257143, 16145599496186285, 94103450048860579

Offset: 0

Kenneth J Ramsey, Apr 17 2012

If p is a prime of the form 8*r +/- 3 then a(p) == 1 (mod p); if p is a prime of the form 8*r +/- 1 then a(p) == 5 (mod p).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A000129, A002203, A038723.

I:=[-1,5]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..19]]; // Bruno Berselli, May 15 2012

m = 19;n = 1; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m + 12; Sow[t];m = n; n = t;c++]][[2,1]]
Table[LucasL[2*n, 2] +Fibonacci[2*n, 2] -3, {n, 0, 40}] (* G. C. Greubel, May 24 2021 *)

[lucas_number1(2*n+2,2,-1) - 2*lucas_number1(2*n,2,-1) -3 for n in (0..40)] # G. C. Greubel, May 24 2021

G.f.: -(1-12*x-x^2)/((1-x)*(1-6*x+x^2)). - Bruno Berselli, May 15 2012
a(n) = 2*A038723(n) - 3. -Bruno Berselli, May 16 2012
a(n) = -3 + (1/4)*( (4+sqrt(2))*(3+2*sqrt(2))^n + (4-sqrt(2))*(3-2*sqrt(2))^n ). - Colin Barker, Mar 05 2016
From G. C. Greubel, May 24 2021: (Start)
a(n) = A000129(2*n+2) - 2*A000129(2*n) - 3.
a(n) = A000129(2*n) + A002203(2*n) - 3. (End)

A266507 a(n) = 6*a(n - 1) - a(n - 2) with a(0) = 2, a(1) = 8.

Original entry on oeis.org

2, 8, 46, 268, 1562, 9104, 53062, 309268, 1802546, 10506008, 61233502, 356895004, 2080136522, 12123924128, 70663408246, 411856525348, 2400475743842, 13990997937704, 81545511882382, 475282073356588, 2770146928257146, 16145599496186288, 94103450048860582

Offset: 0

Raphie Frank, Dec 30 2015

Bisection of A078343 = A078343(2*n + 1).
Quadrisection of A266504 = A266504(4*n + 1).
Octasection of A266506 = A266506(8*n + 2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Bisection of A078343 = A078343(2n + 1).
Quadrisection of A266504 = A266504(4n + 1).
Octasection of A266506 = A266506(8n + 2).
Equals 2*A038723(n).

I:=[2,8]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

LinearRecurrence[{6, -1}, {2, 8}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[2 (1 - 2 x)/(1 - 6 x + x^2), {x, 0, n}], {n, 0, 22}] (* Michael De Vlieger, Dec 31 2015 *)

Vec(2*(1-2*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Dec 31 2015

a(n) = (-sqrt(2)*(1+sqrt(2))^(2*n+1) - 3 *(1-sqrt(2))^(2*n+1) - sqrt(2)*(1-sqrt(2))^(2*n+1) + 3*(1+sqrt(2))^(2*n+1))/sqrt(8).

G.f.: 2*(1-2*x) / (1-6*x+x^2). - Colin Barker, Dec 31 2015

A227792 Expansion of (1 + 6x + 17x^2 - x^3 - 3x^4)/(1 - 6x^2 + x^4).

Original entry on oeis.org

1, 6, 23, 35, 134, 204, 781, 1189, 4552, 6930, 26531, 40391, 154634, 235416, 901273, 1372105, 5253004, 7997214, 30616751, 46611179, 178447502, 271669860, 1040068261, 1583407981, 6061962064, 9228778026, 35331704123, 53789260175, 205928262674

Offset: 0

Ralf Stephan, Sep 23 2013

Also, values i where A067060(i)/i reaches a new maximum (conjectured).

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1)

Cf. A041017.

CoefficientList[Series[(1+6x+17x^2-x^3-3x^4)/(1-6x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,6,0,-1},{1,6,23,35,134},40] (* Harvey P. Dale, Jun 12 2021 *)

a(n)=polcoeff((-3*x^4-x^3+17*x^2+6*x+1)/(x^4-6*x^2+1)+O(x^100),n)

G.f.: (1+6*x+17*x^2-x^3-3*x^4)/((1+2*x-x^2)*(1-2*x-x^2)).
a(2n) = A038723(n+1), n>0.
a(2n+1) = A001109(n+2).
a(n) = (1/4) * (A135532(n+3) + (-1)^n*A001333(n+2) ).

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

A111648 a(n) = A001541(n)^2 + A001653(n+1)^2 + A002315(n)^2.

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A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.

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A266507 a(n) = 6*a(n - 1) - a(n - 2) with a(0) = 2, a(1) = 8.

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A227792 Expansion of (1 + 6*x + 17*x^2 - x^3 - 3*x^4)/(1 - 6*x^2 + x^4).

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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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A227792 Expansion of (1 + 6x + 17x^2 - x^3 - 3x^4)/(1 - 6x^2 + x^4).