A182188 -id:A182188 - cp's OEIS frontend

A182119 Table of triangular arguments such that if A002262(14k) = "r" then the product A182439(k,i + 1) A182439(k,i + 2) equals "r" + A000217(a(k,i)) for i<4, while a(k,i) = 0 for i>3.

0, 55, 4, 384, 51, 7, 2303, 328, 48, 8, 0, 1943, 287, 47, 10, 0, 0, 1680, 276, 45, 11, 0, 0, 0, 1611, 250, 44, 12, 0, 0, 0, 0, 1445, 239, 43, 13, 0, 0, 0, 0, 0, 1376, 228, 42, 14, 0, 0, 0, 0, 0, 1307, 213, 41, 15

Offset: 0

Kenneth J Ramsey, Apr 12 2012

The triangular product A000217(a(k,i)) for i < 4 + A002262(14*k) = the product of adjacent terms G(k,i+1)*G(k,i+2) where G is table A182439. The remainder of each row is padded with zeros. However, if for i > 3, a(k,i) were set to equal 7*a(k,i-1) - 7*a(k,i-2) + a(k,i-3) then the relation above would not be limited to i < 4.
Also, it is noted that the difference between adjacent rows of the respective elements, depends on the difference between the elements of column 0 in the respective rows. In the Mathematica program below, m is set to 14; however, regardless of it value of m, it is apparent that the series of differences corresponding to a difference of d in column 0, i.e. A(k+1,0) - A(k,0) = d, is defined as follows: D(0) = d, D(1) = - d, D(n) = 6*D(n-1) - D(n-2) -8*d + 4. The sequence of differences corresponding to a difference of 1 or 0 in column 0 form related series A182188 and A182190.
The Mathematica program below basically first computes only the nonnegative triangular arguments P. Then it changes at most two of the arguments P in each row k to the corresponding negative value, N = -P -1, in order to obtain the relation a(k,3) = a(k,0) - 7*a(k,1) + 7*a(k,2).

Cf. A182439, A182188, A182190

highTri = Compile[{{S1,_Integer}},Module[{xS0=0,xS1=S1},
While[xS1-xS0*(xS0+1)/2>xS0,xS0++];
xS0]];
overTri = Compile[{{S2,_Integer}},Module[{xS0=0,xS2=S2},
While[xS2-xS0*(xS0+1)/2>xS0,xS0++];
xS2 - (xS0*(1+xS0)/2)]];
tt = SparseArray[{{12,1} -> 1,{1,12} -> 1}];
K1 = 0;
m = 14;While[K1<12,J1=highTri[m*K1];X =2*(m+K1-(J1*2+1));
K2 = 6 m - K1 + X; K3 = 6 K2 - m + X;K4 = 6 K3 - K2 + X;
o = overTri[m*K1]; tt[[1,K1+1]] =highTri[m*K1];
tt[[2,K1+1]] = highTri[m*K2-o];tt[[3,K1+1]] = highTri[K2*K3-o];tt[[4,K1+1]] = highTri[K3*K4-o];
K1++];k = 1;
While[k<13,z = 1; xx = 99; While[z<5 && xx == 99,
If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,Break[]];
If[z == 1,t = -tt[[z,k]]-1;tt[[z,k]] = t,s = -tt[[z-1,k]]-1;tt[[z-1,k]]=s;t =-tt[[z,k]]-1];tt[[z,k]] = t;
w = 1;While[w<5 && xx == 99,If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,xx =0;Break[]];If[w==z,w++];
t=-tt[[w,k]] - 1;tt[[w,k]]=t;If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,xx =0;Break[],
t = -tt[[w,k]] - 1];tt[[w,k]] = t;w++];z++];cc = tt[[1,k]] -6 tt[[2,k]] + tt[[3,k]];p = 5;While[p < 14-k,
tt[[p,k]] = 6 tt[[p-1,k]] - tt[[p-2,k]] + cc;p++]; k++];
a=1;list2 = Reap[While[a<12, b=a; While[b>4,Sow[0];b--];While[b>0, Sow[tt[[b, a+1-b]]]; b--]; a++]][[2, 1]];list2

A182193 Sequence of row differences related to table A182355.

Original entry on oeis.org

-1, 1, 19, 125, 743, 4345, 25339, 147701, 860879, 5017585, 29244643, 170450285, 993457079, 5790292201, 33748296139, 196699484645, 1146448611743, 6681992185825, 38945504503219, 226991034833501, 1323000704497799, 7711013192153305, 44943078448422043

Offset: 0

Kenneth J Ramsey, Apr 17 2012

Sequence of row differences in table A182355. If A182355(k + 1, 0) - A182355(k, 0) = -1, a(n) = A182355(k + 1, n) - A182355(k, n).

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A000129, A002203, A182188, A182189, A182191, A182355.

I:=[-1,1]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..30]]; // Vincenzo Librandi, Feb 10 2014

Pell:= proc(n) option remember;
    if n<2 then n
  else 2*Pell(n-1) + Pell(n-2)
    fi; end:
seq(Pell(2*n) + 2*Pell(2*n-1) - 3, n=0..40); # G. C. Greubel, May 24 2021

LinearRecurrence[{7,-7,1},{-1,1,19},30] (* Harvey P. Dale, Feb 09 2014 *)

Vec(-(1-8*x-5*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 05 2016

[lucas_number2(2*n,2,-1) - lucas_number1(2*n,2,-1) - 3 for n in (0..40)] # G. C. Greubel, May 24 2021

a(n) = 6*a(n-1) - a(n-2) + 12.
a(0)=-1, a(1)=1, a(2)=19, a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - Harvey P. Dale, Feb 09 2014
From Colin Barker, Mar 05 2016: (Start)
a(n) = -3 + (1/4)*( (4-sqrt(2))*(3+2*sqrt(2))^n + (4+sqrt(2))*(3-2*sqrt(2))^n ).
G.f.: -(1-8*x-5*x^2) / ((1-x)*(1-6*x+x^2)).
(End)
a(n) = A002203(2*n) - A000129(2*n) - 3. - G. C. Greubel, May 24 2021

More terms from Harvey P. Dale, Feb 09 2014

A182355 Table of triangular arguments such that if A002262(14k) = "r" then the product A182441(k,i + 1) A182441(k,i + 2) equals "r" + a(k,i)*(a(k,i) + 1)/2 for i<4, while a(k,i) = 0 for i>3.

Original entry on oeis.org

-1, 56, -5, 399, 60, -8, 2400, 463, 63, -9, 0, 2816, 512, 64, -11, 0, 0, 3135, 531, 66, -12, 0, 0, 0, 3260, 565, 67, -13, 0, 0, 0, 0, 3482, 584, 68, -14, 0, 0, 0, 0, 0, 3607, 603, 69, -15, 0, 0, 0, 0, 0, 0, 3732, 622

Offset: 0

Kenneth J Ramsey, Apr 25 2012

The triangular product a(k,i)*(a(k,i)+1)/2 + A002262(14*k) for i<4 = the product of adjacent terms G(k,i+1)*G(k,i+2) where G is table A182441. The remainder of each row is padded with zeros. However, if for i > 3, a(k,i) were set to equal 7*a(k,i-1) - 7*a(k,i-2) + a(k,i-3) then the relation above would not be limited to i < 4.
Also, it is noted that the difference between adjacent rows of the respective elements, depends on the difference between the elements of column 0 in the respective rows. In the Mathematica program below, m is set to 14; however, regardless of it value of m, it is apparent that the series of differences corresponding to a difference of d in column 0, i.e. A(k+1,0) - A(k,0) = d, is defined as follows: D(0) = d, D(1) = - d, D(n) = 6*D(n-1) - D(n-2) -8*d + 4. The sequence of differences corresponding to a difference d of -1 is series A182193.
The Mathematica program below basically first computes only the nonnegative triangular arguments P. Then it changes at most two of the arguments P in each row k to the corresponding negative value, N = -P -1, in order to obtain the relation a(k,3) = a(k,0) - 7*a(k,1) + 7*a(k,2).

Cf. A182102, A182118, A182119, A182190, A182193, A182188, A182189.

highTri = Compile[{{S1,_Integer}},Module[{xS0=0,xS1=S1},
While[xS1-xS0*(xS0+1)/2>xS0,xS0++];
xS0]];
overTri = Compile[{{S2,_Integer}},Module[{xS0=0,xS2=S2},
While[xS2-xS0*(xS0+1)/2>xS0,xS0++];
xS2 - (xS0*(1+xS0)/2)]];
tt = SparseArray[{{12,1} -> 0,{1,12} -> 0}];
K1 = 0;
m = 14;While[K1<12,J1=highTri[m*K1];X =2*(m+K1+(J1*2+1));
K2 = 6 m - K1 + X; K3 = 6 K2 - m + X;K4 = 6 K3 - K2 + X;
o = overTri[m*K1]; tt[[1,K1+1]] =highTri[m*K1];
tt[[2,K1+1]] = highTri[m*K2-o];tt[[3,K1+1]] = highTri[K2*K3-o];tt[[4,K1+1]] = highTri[K3*K4-o];
K1++];k = 1;
While[k<13,z = 1; xx = 99; While[z<5 && xx == 99,
If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,Break[]];
If[z == 1,t = -tt[[z,k]]-1;tt[[z,k]] = t,s = -tt[[z-1,k]]-1;tt[[z-1,k]]=s;t =-tt[[z,k]]-1];tt[[z,k]] = t;
w = 1;While[w<5 && xx == 99,If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,xx =0;Break[]];If[w==z,w++];
t=-tt[[w,k]] - 1;tt[[w,k]]=t;If[tt[[1,k]]+ 7 tt[[3,k]] - 7 tt[[2,k]] - tt[[4,k]] == 0,xx =0;Break[],
t = -tt[[w,k]] - 1];tt[[w,k]] = t;w++];z++];cc = tt[[1,k]] -6 tt[[2,k]] + tt[[3,k]];p = 5;While[p < 14-k,
tt[[p,k]] = 6 tt[[p-1,k]] - tt[[p-2,k]] + cc;p++]; k++];
a=1;list2 = Reap[While[a<12, b=a; While[b>4,Sow[0];b--];While[b>0, Sow[tt[[b, a+1-b]]]; b--]; a++]][[2, 1]];list2

cp's OEIS Frontend

A182119 Table of triangular arguments such that if A002262(14k) = "r" then the product A182439(k,i + 1) A182439(k,i + 2) equals "r" + A000217(a(k,i)) for i<4, while a(k,i) = 0 for i>3.

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A182193 Sequence of row differences related to table A182355.

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A182355 Table of triangular arguments such that if A002262(14k) = "r" then the product A182441(k,i + 1) A182441(k,i + 2) equals "r" + a(k,i)*(a(k,i) + 1)/2 for i<4, while a(k,i) = 0 for i>3.

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cp's OEIS Frontend

A182119 Table of triangular arguments such that if A002262(14*k) = "r" then the product A182439(k,i + 1) *A182439(k,i + 2) equals "r" + A000217(a(k,i)) for i<4, while a(k,i) = 0 for i>3.

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Programs

Mathematica

A182193 Sequence of row differences related to table A182355.

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Author

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Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A182355 Table of triangular arguments such that if A002262(14*k) = "r" then the product A182441(k,i + 1) *A182441(k,i + 2) equals "r" + a(k,i)*(a(k,i) + 1)/2 for i<4, while a(k,i) = 0 for i>3.

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Mathematica

A182119 Table of triangular arguments such that if A002262(14k) = "r" then the product A182439(k,i + 1) A182439(k,i + 2) equals "r" + A000217(a(k,i)) for i<4, while a(k,i) = 0 for i>3.

A182355 Table of triangular arguments such that if A002262(14k) = "r" then the product A182441(k,i + 1) A182441(k,i + 2) equals "r" + a(k,i)*(a(k,i) + 1)/2 for i<4, while a(k,i) = 0 for i>3.