A147703 -id:A147703 - cp's OEIS frontend

A147837 a(n)=7a(n-1)-5a(n-2), a(0)=1, a(1)=5.

1, 5, 30, 185, 1145, 7090, 43905, 271885, 1683670, 10426265, 64565505, 399827210, 2475962945, 15332604565, 94948417230, 587975897785, 3641089198345, 22547744899490, 139628768304705, 864662653635485, 5354494733924870, 33158149869296665, 205334575415452305

Offset: 0

Philippe Deléham, Nov 14 2008

Index entries for linear recurrences with constant coefficients, signature (7,-5).

Cf. A147703.

LinearRecurrence[{7,-5},{1,5},30] (* Harvey P. Dale, Oct 10 2015 *)

a(n) = Sum_{k=0..n} A147703(n,k)*4^k.
G.f.: (1-2*x)/(1-7*x+5*x^2).
a(n) = ((29+3*sqrt(29))/58)*(3.5+0.5*sqrt(29))^n +((29-3*sqrt(29))/58)*(3.5-0.5*sqrt(29))^n. - Richard Choulet, Nov 20 2008 [corrected by Jason Yuen, Oct 04 2024]

A147839 a(n)=9a(n-1)-7a(n-2), a(0)=1, a(1)=7 .

Original entry on oeis.org

1, 7, 56, 455, 3703, 30142, 245357, 1997219, 16257472, 132336715, 1077228131, 8768696174, 71377668649, 581018144623, 4729519621064, 38498549577215, 313380308847487, 2550932932586878, 20764734231349493, 169026077554037291

Offset: 0

Philippe Deléham, Nov 14 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (9,-7).

LinearRecurrence[{9,-7},{1,7},30] (* Harvey P. Dale, Nov 30 2019 *)

a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*6^k . G.f.: (1-2x)/(1-9x+7*x^2).

a(n)= ((53+5*sqrt(53))/106)*(4.5+0.5*sqrt(53))^n + ((53-5*sqrt(53))/106)*(4.5-0.5*sqrt(53))^n [From Richard Choulet, Nov 20 2008]

A147840 a(n)=10a(n-1)-8a(n-2), a(0)=1, a(1)=8 .

Original entry on oeis.org

1, 8, 72, 656, 5984, 54592, 498048, 4543744, 41453056, 378180608, 3450181632, 31476371456, 287162261504, 2619811643392, 23900818341888, 218049690271744, 1989290355982336, 18148506037649408, 165570737528635392

Offset: 0

Philippe Deléham, Nov 14 2008

a(n) = sum_{k=0..n} 2^n*binomial(n,k)*A007482(k) = 2^n*A052913(n). - R. J. Mathar, Oct 15 2012

Index entries for linear recurrences with constant coefficients, signature (10,-8).

LinearRecurrence[{10,-8},{1,8},20] (* Harvey P. Dale, Dec 02 2021 *)

a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*7^k . G.f.: (1-2x)/(1-10x+8*x^2).
a(n)= ((17+3*sqrt(17))/34)*(5+sqrt(17))^n + ((17-3*sqrt(17))/34)*(5-sqrt(17))^n [From Richard Choulet, Nov 20 2008]
G.f.: (1-2x)/(1-10x+8x^2). - Harvey P. Dale, Dec 02 2021

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

Original entry on oeis.org

1, 4, 28, 232, 1984, 17056, 146752, 1262848, 10867456, 93520384, 804794368, 6925699072, 59599458304, 512886194176, 4413668442112, 37982050091008, 326856479604736, 2812780194955264, 24205524194295808, 208301879603494912, 1792552505703399424, 15425902501792055296

Offset: 0

Philippe Deléham, Dec 09 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-12).

Cf. A052961, A006012, A147703, A152596.

LinearRecurrence[{10, -12}, {1, 4}, 25] (* Paolo Xausa, Jan 19 2024 *)

G.f.: (1-6*x)/(1-10*x+12*x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*3^(n-k).
a(n) = 2^n*A052961(n). - R. J. Mathar, Jun 14 2016

A152620 a(n)=-8a(n-1)-6a(n-2), n>1 ; a(0)=1, a(1)=-2 .

Original entry on oeis.org

1, -2, 10, -68, 484, -3464, 24808, -177680, 1272592, -9114656, 65281696, -467565632, 3348834880, -23985285248, 171789272704, -1230402470144, 8812484124928, -63117458178560, 452064760678912, -3237813336359936

Offset: 0

Philippe Deléham, Dec 10 2008

sign

Index entries for linear recurrences with constant coefficients, signature (-8,-6).

Cf. A152594

LinearRecurrence[{-8,-6},{1,-2},30] (* Harvey P. Dale, Apr 03 2015 *)

G.f.: (1+6*x)/(1+8*x+6*x^2). a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*(-3)^(n-k).

A206831 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, -1, -3, -1, 1, 1, 0, -4, -2, 1, 1, 5, 4, -4, -3, 1, -1, 0, 9, 10, -3, -4, 1, -1, -7, -9, 9, 17, -1, -5, 1, 1, 0, -16, -28, 2, 24, 2, -6, 1, 1, 9, 16, -16, -54, -14, 30, 6, -7, 1, -1, 0, 25, 60, 10

Offset: 0

Philippe Deléham, Feb 13 2012

Riordan array ((1+x)/(1+x^2), x*(1-x)/(1+x^2)).

Antidiagonal sums are A010892(n).

			Triangle begins :
1
1, 1
-1, 0, 1
-1, -3, -1, 1
1, 0, -4, -2, 1
1, 5, 4, -4, -3, 1
-1, 0, 9, 10, -3, -4, 1
-1, -7, -9, 9, 17, -1, -5, 1
1, 0, -16, -28, 2, 24, 2, -6, 1
1, 9, 16, -16, -54, -14, 30, 6, -7, 1
-1, 0, 25, 60, 10, -80, -40, 34, 11, -8, 1

Indranil Ghosh, Rows 0..100, flattened

Cf. A129267, A007318, A147703, A147721

Cf. A057077, A087960

nmax=10; Flatten[CoefficientList[Series[CoefficientList[Series[(1 + x)/(1 - y*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

T(n,k) = T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), n>1.
G.f.: (1+x)/(1-y*x+(1+y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A057077(n), (-1)^n*A078050(n) for x = -1, 0, 1 respectively.

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 27, 13, 1, 9, 35, 73, 80, 34, 1, 11, 54, 151, 252, 234, 89, 1, 13, 77, 269, 597, 837, 677, 233, 1, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 1, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597

Offset: 0

Philippe Deléham, Nov 06 2011

Mirror image of triangle in A147703.

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,  5;
  1,  7, 20, 27, 13;
  1,  9, 35, 73, 80, 34;

Cf. A001519, A059502, A000012, A005408, A014107.

Sum_{k=0..n} T(n,k)*x^k = A152620(n), A152594(n), A000007(n), A000012(n), A006012(n), A152596(n), A152599(n) for x=-3,-2,-1,0,1,2,3 respectively.
T(n,n) = A001519(n).
G.f.: (1-2y*x)/(1-(1+3y)*x+y*(1+y)*x^2).

cp's OEIS Frontend

A147837 a(n)=7*a(n-1)-5*a(n-2), a(0)=1, a(1)=5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A147839 a(n)=9*a(n-1)-7*a(n-2), a(0)=1, a(1)=7 .

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A147840 a(n)=10*a(n-1)-8*a(n-2), a(0)=1, a(1)=8 .

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A152599 a(n) = 10*a(n-1) - 12*a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A152620 a(n)=-8*a(n-1)-6*a(n-2), n>1 ; a(0)=1, a(1)=-2 .

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A206831 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A147837 a(n)=7a(n-1)-5a(n-2), a(0)=1, a(1)=5.

A147839 a(n)=9a(n-1)-7a(n-2), a(0)=1, a(1)=7 .

A147840 a(n)=10a(n-1)-8a(n-2), a(0)=1, a(1)=8 .

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

A152620 a(n)=-8a(n-1)-6a(n-2), n>1 ; a(0)=1, a(1)=-2 .