A002114 -id:A002114 - cp's OEIS frontend

A186433 Matrix inverse of A186432.

1, -1, 1, 11, -12, 1, -301, 330, -30, 1, 15371, -16856, 1540, -56, 1, -1261501, 1383390, -126420, 4620, -90, 1, 151846331, -166518132, 15217290, -556248, 10890, -132, 1, -25201039501, 27636032242, -2525525002, 92318226, -1807806, 22022, -182, 1

Offset: 0

Peter Bala, Feb 22 2011

			Triangle begins
n/k.|.........0...........1.........2........3.......4......5.....6
===================================================================
.0..|.........1
.1..|........-1...........1
.2..|........11.........-12.........1
.3..|......-301.........330.......-30........1
.4..|.....15371......-16856......1540......-56.......1
.5..|..-1261501.....1383390...-126420.....4620.....-90......1
.6..|.151846331..-166518132..15217290..-556248...10890...-132.....1
..

A002114, A186432 (inverse).

GENERATING FUNCTION
Conjectural e.g.f.:
... 1/2+1/2{(2*cosh(sqrt(u)*z)-1)/(2*cosh(z)-1)}
= sum {n = 0..inf} R(n,u)*z^(2*n)/(2*n)!
= 1+(u-1)*z^2/2!+(u^2-12*u+11)*z^4/4!+....
RELATIONS WITH OTHER SEQUENCES
Column 0: Signed version of Glaisher's H' numbers A002114.

A226276 Period 4: repeat [8, 4, 4, 4].

Original entry on oeis.org

8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4, 8, 4, 4, 4

Offset: 0

Richard R. Forberg, Jun 01 2013

Old name was: A four-term repeating sequence for constructing a summation sequence from negative to positive infinity containing all primes except 2 and 5.
a(n) allows for the creation of an infinite summation sequence, s(n), extending from negative to positive infinity. (See Formula section below.) With appropriate initialization, letting "s(n+)" be the set positive s(n) values, and "s(n-)" be the absolute value of the set of negative s(n) values, the following applies:
s(n+) includes all primes of the form 4*m+1 with m>=2. Thus excluding 5.
s(n-) includes all primes of the form 4*m+3 with m>=0.
Together these include all primes (except 2 and 5) without duplication.
The primes "p(+)" within s(n+) "appear" in the form 3*p(+) within s(n-).
The primes "p(-)" within s(n-) "appear" in the form 3*p(-) within s(n+).
By using this simple repeating pattern, rather than the two well known linear formulas above, all primes (except 2 and 5) are included via a single construction mechanism, and all integers ending in the digit 5 are excluded mathematically, resulting in fewer nonprimes among the values of s(n) than there are in the combination of 4*m+1 and 4*m+3.
(NOTE: In the above "m" is not that same index as "n").
This is one of only two such repeating sequences with the property of generating a summation sequence that includes all integers ending in 1,3,7 or 9, and thus all primes except 2 and 5 (for the other see A226294). Both have the same density of primes in s(n), because both generate only 40% of the integers (in absolute value). And both presumably have the same average density of primes in positive vs. negative values of s(n).
Also, continued fraction expansion of 4 + sqrt(646)/6. - Bruno Berselli, Jun 20 2013

			s(1) = 9, s(2) = 13, s(3) = 17, s(4) = 21, s(5) = 29, s(6) = 33, s(7) = 37.
s(-1) = -3, s(-2) = -7, s(-3) = -11, s(-4) = -19, s(-5) = -23, s(-6) = -27, s(-7) = -31.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A002114, A002145, A121262, A177704, A226294.

&cat [[8, 4, 4, 4]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([8, 4, 4, 4]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

Flatten[Table[{8, 4, 4, 4}, {20}]] (* Bruno Berselli, Jun 20 2013 *)
PadRight[{},120,{8,4,4,4}] (* Harvey P. Dale, May 11 2025 *)

a(n)=if(n%4,4,8) \\ Charles R Greathouse IV, Jul 17 2016

For generating the summation sequence s, start with s(0) = 1, and a(0) = 8.
For positive values of s(n): s(n+1) = s(n) + a(n).
For negative values of s(n): s(n-1) = s(n) - a(n-1). Here, n is negative.
All values of a(n) are positive regardless of index. For example: a(-1) = a(-2) = a(-3) = 4; a(-4) = 8. Thus the simple pattern of a(n) and the simple arithmetic for generating s(n), are maintained across the n=0 boundary, in a manner similar to extending Fibonacci numbers to negative indices.
From Bruno Berselli, Jun 20 2013: (Start)
G.f.: 4*(2+x+x^2+x^3)/((1-x)*(1+x)*(1+x^2)).
a(n) = 4 + (1 + (-1)^n)*(1 + I^(n*(n+1))). (End)
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = 5 + I^(2*n) + I^(-n) + I^n.
a(n) = 5 + cos(n*Pi) + 2*cos(n*Pi/2) + I*sin(n*Pi). (End)

Simpler name from Joerg Arndt, Jun 16 2013

A331612 E.g.f.: exp(1 / (2 - sec(x)) - 1) (even powers only).

Original entry on oeis.org

1, 1, 14, 481, 30449, 3064306, 448104029, 89621046061, 23468873468054, 7786478152466221, 3190021872763911149, 1580829351026679822586, 931656913226081002622489, 643808850722810399312420281, 515431991397502094847830786174, 473171296200788822261644150349881

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A000364, A002114, A217502, A331611.

nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Sec[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
e[0] = 1; e[n_] := e[n] = (-1)^n (1 - Sum[(-1)^j Binomial[2 n, 2 j] 3^(2 (n - j)) e[j], {j, 0, n - 1}]); A002114[n_] := e[n]/2^(2 n + 1); a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A002114[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
With[{nn=40},Take[CoefficientList[Series[Exp[1/(2-Sec[x])-1],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 08 2023 *)

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * A002114(k) * a(n-k).

a(n) ~ 2^(2*n) * 3^(2*n + 1/8) * exp(-5/12 + sqrt(3)/(4*Pi) + 2*3^(1/4)*sqrt(n/Pi) - 2*n) * n^(2*n - 1/4) / Pi^(2*n + 1/4). - Vaclav Kotesovec, Jan 26 2020

A161665 Primes that can be represented as a sum of 2 and also as a sum of 3 distinct nonzero squares, sharing a term in the sums.

Original entry on oeis.org

29, 101, 109, 149, 173, 181, 229, 233, 241, 269, 293, 389, 401, 409, 421, 433, 449, 521, 569, 641, 661, 677, 701, 757, 761, 769, 797, 821, 857, 877, 881, 941, 1021, 1069, 1097, 1109, 1117, 1181, 1229, 1237, 1277, 1289, 1301, 1373, 1381, 1429, 1433, 1481, 1549

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 15 2009

nonn

Dropping the requirement of one shared term, we would get the supersequence 17, 29, 41, 53, 61, 73, ... - R. J. Mathar, Oct 04 2009

			The prime 29 has the representations 29 = 2^2+ 5^2 = 2^2+3^2+4^2, sharing 2^2.
The prime 101 has the representations 101 = 1^2+10^2 = 1^2+6^2+8^2, sharing 1^2.
The prime 109 has the representations 109 = 3^2+10^2 = 3^2+6^2+8^2, sharing 3^2.
The prime 149 has the representations 149 = 7^2+10^2 = 6^2+7^2+8^2, sharing 7^2.

Cf. A002114, A085317.

f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[a=f[n];If[a>0,b=f[n-(f[n])^2]; If[b>0,c=(n-a^2-b^2)^(1/2);If[a!=b&&a!=c,If[PrimeQ[n],AppendTo[lst, n]]]]],{n,3,4*6!}];lst

Definition reverse-engineered from program by R. J. Mathar, Oct 04 2009

cp's OEIS Frontend

A186433 Matrix inverse of A186432.

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A226276 Period 4: repeat [8, 4, 4, 4].

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A331612 E.g.f.: exp(1 / (2 - sec(x)) - 1) (even powers only).

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A161665 Primes that can be represented as a sum of 2 and also as a sum of 3 distinct nonzero squares, sharing a term in the sums.

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Mathematica

Extensions