A017473 -id:A017473 - cp's OEIS frontend

A254963 a(n) = n(11n + 3)/2.

0, 7, 25, 54, 94, 145, 207, 280, 364, 459, 565, 682, 810, 949, 1099, 1260, 1432, 1615, 1809, 2014, 2230, 2457, 2695, 2944, 3204, 3475, 3757, 4050, 4354, 4669, 4995, 5332, 5680, 6039, 6409, 6790, 7182, 7585, 7999, 8424, 8860, 9307, 9765, 10234, 10714, 11205, 11707

Offset: 0

Bruno Berselli, Feb 11 2015

This sequence provides the first differences of A254407 and the partial sums of A017473.
Also:
a(n) - n         = A022269(n);
a(n) + n         = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;
a(n) - 2*n       = A022268(n);
a(n) + 2*n       = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;
a(n) - 3*n       = n*(11*n-3)/2: 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) + 3*n       = A211013(n);
a(n) - 4*n       = A226492(n);
a(n) + 4*n       = A152740(n);
a(n) - 5*n       = A180223(n);
a(n) + 5*n       = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;
a(n) - 6*n       = A051865(n);
a(n) + 6*n       = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;
a(n) - 7*n       = A152740(n-1) with A152740(-1) = 0;
a(n) + 7*n       = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;
a(n) - n*(n-1)/2 = A168668(n);
a(n) + n*(n-1)/2 = A049453(n);
a(n) - n*(n+1)/2 = A202803(n);
a(n) + n*(n+1)/2 = A033580(n).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008729 and A218530 (seventh column); A017473, A254407.
Cf. similar sequences of the type 4*n^2 + k*n*(n+1)/2: A055999 (k=-7, n>6), A028552 (k=-6, n>2), A095794 (k=-5, n>1), A046092 (k=-4, n>0), A000566 (k=-3), A049450 (k=-2), A022264 (k=-1), A016742 (k=0), A022267 (k=1), A202803 (k=2), this sequence (k=3), A033580 (k=4).
Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.
Cf. A022268, A022269, A049453, A051865, A152740, A168668, A180223, A211013, A226492.

Magma
```
[n*(11*n+3)/2: n in [0..50]];
```

Table[n (11 n + 3)/2, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,7,25},50] (* Harvey P. Dale, Mar 25 2018 *)

Maxima
```
makelist(n*(11*n+3)/2, n, 0, 50);
```
PARI
```
vector(50, n, n--; n*(11*n+3)/2)
```
Sage
```
[n*(11*n+3)/2 for n in (0..50)]
```

G.f.: x*(7 + 4*x)/(1 - x)^3.
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(14 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A017474 a(n) = (11*n + 7)^2.

Original entry on oeis.org

49, 324, 841, 1600, 2601, 3844, 5329, 7056, 9025, 11236, 13689, 16384, 19321, 22500, 25921, 29584, 33489, 37636, 42025, 46656, 51529, 56644, 62001, 67600, 73441, 79524, 85849, 92416, 99225, 106276, 113569, 121104, 128881, 136900, 145161, 153664, 162409, 171396, 180625, 190096, 199809

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Powers of the form (11*n+7)^m: A017473 (m=1), this sequence (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..50], n-> (11*n+7)^2); # G. C. Greubel, Sep 19 2019

[(11*n+7)^2: n in [0..50]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^2, n=0..50); # G. C. Greubel, Sep 19 2019

(11 Range[0,50]+7)^2 (* or *) LinearRecurrence[{3,-3,1},{49,324,841},50] (* Harvey P. Dale, May 19 2019 *)

a(n)=(11*n+7)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(11*n+7)^2 for n in (0..50)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (49 +177*x +16*x^2)/(1-x)^3.
E.g.f.: (49 +275*x +121*x^2)*exp(x). (End)

A017475 a(n) = (11*n + 7)^3.

Original entry on oeis.org

343, 5832, 24389, 64000, 132651, 238328, 389017, 592704, 857375, 1191016, 1601613, 2097152, 2685619, 3375000, 4173281, 5088448, 6128487, 7301384, 8615125, 10077696, 11697083, 13481272, 15438249, 17576000, 19902511, 22425768, 25153757, 28094464, 31255875, 34645976, 38272753, 42144192

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), this sequence (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..40], n-> (11*n+7)^3); # G. C. Greubel, Sep 19 2019

[(11*n+7)^3: n in [0..40]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^3, n=0..40); # G. C. Greubel, Sep 19 2019

(11*Range[0,40]+7)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{343,5832, 24389,64000}, 40] (* Harvey P. Dale, Oct 18 2014 *)

makelist((11*n+7)^3, n,0,40); /* Martin Ettl, Oct 21 2012 */

a(n) = (11*n+7)^3; \\ Altug Alkan, Sep 08 2018

[(11*n+7)^3 for n in (0..40)] # G. C. Greubel, Sep 19 2019

G.f.: (343 + 4460*x + 3119*x^2 + 64*x^3)/(1-x)^4. - R. J. Mathar, Jun 24 2009
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=343, a(1)=5832, a(2)=24389, a(3)=64000. - Harvey P. Dale, Oct 18 2014
E.g.f.: (343 +5489*x +6534*x^2 +1331*x^3)*exp(x). - G. C. Greubel, Sep 19 2019

A017476 a(n) = (11*n + 7)^4.

Original entry on oeis.org

2401, 104976, 707281, 2560000, 6765201, 14776336, 28398241, 49787136, 81450625, 126247696, 187388721, 268435456, 373301041, 506250000, 671898241, 875213056, 1121513121, 1416468496, 1766100625, 2176782336, 2655237841, 3208542736, 3844124001, 4569760000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), this sequence (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..30], n-> (11*n+7)^4); # G. C. Greubel, Sep 19 2019

[(11*n+7)^4: n in [0..30]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^4, n=0..30); # G. C. Greubel, Sep 19 2019

(11*Range[0,30]+7)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1}, {2401, 104976,707281,2560000,6765201}, 30] (* Harvey P. Dale, Oct 21 2015 *)

vector(30, n, (11*n-4)^4) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^4 for n in (0..30)] # G. C. Greubel, Sep 19 2019

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=2401, a(1)=104976, a(2)=707281, a(3)=2560000, a(4)=6765201. - Harvey P. Dale, Oct 21 2015
From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (2401 +92971*x +206411*x^2 +49345*x^3 +256*x^4)/(1-x)^5.
E.g.f.: (2401 +102575*x +249865*x^2 +125114*x^3 +14641 x^4)*exp(x). (End)

A017477 a(n) = (11*n + 7)^5.

Original entry on oeis.org

16807, 1889568, 20511149, 102400000, 345025251, 916132832, 2073071593, 4182119424, 7737809375, 13382255776, 21924480357, 34359738368, 51888844699, 75937500000, 108175616801, 150536645632, 205236901143

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), this sequence (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..30], n-> (11*n+7)^5); # G. C. Greubel, Sep 19 2019

[(11*n+7)^5: n in [0..30]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^5, n=0..30); # G. C. Greubel, Sep 19 2019

(11 * Range[0, 30] + 7)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {16807, 1889568, 20511149, 102400000, 345025251, 916132832}, 30] (* Harvey P. Dale, Jan 16 2013 *)

vector(30, n, (11*n-4)^5) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^5 for n in (0..30)] # G. C. Greubel, Sep 19 2019

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), with a(0) = 16807, a(1) = 1889568, a(2) = 20511149, a(3) = 102400000, a(4) = 345025251, a(5) = 916132832. - Harvey P. Dale, Jan 16 2013
From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (16807 +1788726*x +9425846*x^2 +7340486*x^3 +753231*x^4 +1024*x^5 )/(1-x)^6.
E.g.f.: (16807 +1872761*x +8374410*x^2 +7753075*x^3 +2122945*x^4 +161051 *x^5)*exp(x). (End)

A017478 a(n) = (11*n + 7)^6.

Original entry on oeis.org

117649, 34012224, 594823321, 4096000000, 17596287801, 56800235584, 151334226289, 351298031616, 735091890625, 1418519112256, 2565164201769, 4398046511104, 7212549413161, 11390625000000, 17416274304961, 25892303048704, 37558352909169, 53310208315456, 74220378765625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), this sequence (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^6); # G. C. Greubel, Sep 19 2019

[(11*n+7)^6: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^6, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[21] -4)^6 (* G. C. Greubel, Sep 19 2019 *)

makelist((11*n+7)^6, n, 0, 20); /* Martin Ettl, Oct 21 2012 */

vector(20, n, (11*n-4)^6) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^6 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (117649 +33188681*x +359208382*x^2 +642375742*x^3 +229267417*x^4 +11361953*x^5 +4096*x^6)/(1-x)^7.
E.g.f.: (117649 +33894575*x +263458261*x^2 +402241510*x^3 +193554020*x^4 +33337557*x^5 +1771561*x^6)*exp(x). (End)

A017479 a(n) = (11*n + 7)^7.

Original entry on oeis.org

823543, 612220032, 17249876309, 163840000000, 897410677851, 3521614606208, 11047398519097, 29509034655744, 69833729609375, 150363025899136, 300124211606973, 562949953421312, 1002544368429379, 1708593750000000, 2804020163098721, 4453476124377088, 6873178582377927

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), this sequence (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^7); # G. C. Greubel, Sep 19 2019

[(11*n+7)^7: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^7, n=0..20); # G. C. Greubel, Sep 19 2019

(11Range[0,20]+7)^7  (* Harvey P. Dale, Mar 24 2011 *)

vector(20, n, (11*n-4)^7) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^7 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (823543 +605631688*x +12375175257*x^2 +42937032016*x^3 +35460540721 *x^4 +6665393928*x^5 +170728303*x^6 +16384*x^7)/(1-x)^8.
E.g.f.: (823543 +611396489*x +8013129894*x^2 +18987701271*x^3 + 14295911630*x^4 +4196022754*x^5 +496037080*x^6 +19487171*x^7)*exp(x). (End)

A017480 a(n) = (11*n + 7)^8.

Original entry on oeis.org

5764801, 11019960576, 500246412961, 6553600000000, 45767944570401, 218340105584896, 806460091894081, 2478758911082496, 6634204312890625, 15938480745308416, 35114532758015841, 72057594037927936, 139353667211683681, 256289062500000000, 451447246258894081

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), this sequence (m=8), A017481 (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^8); # G. C. Greubel, Sep 19 2019

[(11*n+7)^8: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^8, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[0,20]+7)^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36, -9,1}, {5764801,11019960576, 500246412961,6553600000000, 45767944570401, 218340105584896,806460091894081,2478758911082496,6634204312890625}, 20] (* Harvey P. Dale, Mar 30 2016 *)

vector(20, n, (11*n-4)^8) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^8 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (5764801 +10968077367*x +401274300613*x^2 +2447616620803*x^3 + 3869465113539*x^4 +1725294430213*x^5 +185763408247*x^6 +2562300801*x^7 + 65536*x^8)/(1-x)^9.
E.g.f.: (5764801 +11014195775*x +239106128305*x^2 +847652479674*x^3 + 937956207111*x^4 +417408438678*x^5 +82366957134*x^6 +7093330244*x^7 + 214358881*x^8)*exp(x). (End)

A017481 a(n) = (11*n + 7)^9.

Original entry on oeis.org

40353607, 198359290368, 14507145975869, 262144000000000, 2334165173090451, 13537086546263552, 58871586708267913, 208215748530929664, 630249409724609375, 1689478959002692096, 4108400332687853397, 9223372036854775808, 19370159742424031659, 38443359375000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), this sequence (m=9), A017482 (m=10), A017483 (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^9); # G. C. Greubel, Sep 19 2019

[(11*n+7)^9: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^9, n=0..20); # G. C. Greubel, Sep 19 2019

(11Range[0,20]+7)^9  (* Harvey P. Dale, Mar 23 2011 *)

makelist((11*n+7)^9, n, 0, 30); /* Martin Ettl, Oct 21 2012 */

a(n) = (11*n+7)^9; \\ Altug Alkan, Sep 08 2018

[(11*n+7)^9 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (40353607 +197955754298*x +12525368984504*x^2 +125993865875030*x^3 +341752101417866*x^4 +292702580123078*x^5 +77396622719912*x^6 + 5045081881706*x^7 +38440737935*x^8 +262144*x^9)/(1-x)^10.
E.g.f.: (40353607 +198318936761*x +7055233874370*x^2 +36536266598315*x^3 +57159943839075*x^4 +36196841476257*x^5 +10604280696240*x^6 + 1501876268970*x^7 +98390726379*x^8 +2357947691*x^9)*exp(x). (End)

A017482 a(n) = (11*n + 7)^10.

Original entry on oeis.org

282475249, 3570467226624, 420707233300201, 10485760000000000, 119042423827613001, 839299365868340224, 4297625829703557649, 17490122876598091776, 59873693923837890625, 179084769654285362176, 480682838924478847449, 1180591620717411303424, 2692452204196940400601

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), this sequence (m=10), A017483 (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^10); # G. C. Greubel, Sep 19 2019

[(11*n+7)^10: n in [0..10]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^10, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[21] -4)^10 (* G. C. Greubel, Sep 19 2019 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{282475249,3570467226624,420707233300201,10485760000000000,119042423827613001,839299365868340224,4297625829703557649,17490122876598091776,59873693923837890625,179084769654285362176,480682838924478847449},30] (* Harvey P. Dale, Apr 21 2020 *)

vector(20, n, (11*n-4)^10) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^10 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (282475249 +3567359998885*x +381447629946032*x^2 +6054309522746024* x^3 +26248927783563266*x^4 +38310933951284930*x^5 +19699677304461320*x^6 +3287461918700048*x^7 +134823999028181*x^8 +576638856289*x^9 +1048576* x^10)/(1-x)^11.
E.g.f.: (282475249 +3570184751375*x +206783290661101*x^2 + 1539058236550670*x^3 +3317056068374290*x^4 +2872963553757759*x^5 +1172277747064347*x^6 +242804694252120 x^7 +25867757964675*x^8 +1332240445415*x^9 +25937424601*x^10)*exp(x). (End)

cp's OEIS Frontend

A254963 a(n) = n*(11*n + 3)/2.

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A017474 a(n) = (11*n + 7)^2.

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A017475 a(n) = (11*n + 7)^3.

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GAP

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A017476 a(n) = (11*n + 7)^4.

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Magma

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Sage

Formula

A017477 a(n) = (11*n + 7)^5.

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GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A017478 a(n) = (11*n + 7)^6.

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GAP

A254963 a(n) = n(11n + 3)/2.