A017461 -id:A017461 - cp's OEIS frontend

A269044 a(n) = 13*n + 7.

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Bruno Berselli, Feb 18 2016

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Magma
```
[13*n+7: n in [0..60]];
```

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima
```
makelist(13*n+7, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+7)
```
Sage
```
[13*n+7 for n in (0..60)]
```

G.f.: (7 + 6*x)/(1 - x)^2.
a(n) =  A088227(4*n+3).
a(n) = -A186113(-n-1).
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021

A017462 a(n) = (11*n + 6)^2.

Original entry on oeis.org

36, 289, 784, 1521, 2500, 3721, 5184, 6889, 8836, 11025, 13456, 16129, 19044, 22201, 25600, 29241, 33124, 37249, 41616, 46225, 51076, 56169, 61504, 67081, 72900, 78961, 85264, 91809, 98596, 105625, 112896, 120409, 128164, 136161, 144400, 152881, 161604, 170569

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+6)^m: A017461 (m=1), this sequence (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..50], n-> (11*n+6)^2); # G. C. Greubel, Sep 19 2019

[(11*n+6)^2: n in [0..40]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+6)^2, n=0..50); # G. C. Greubel, Sep 19 2019

(11*Range[0,30]+6)^2 (* or *) LinearRecurrence[{3,-3,1},{36,289,784},30] (* Harvey P. Dale, Sep 24 2016 *)

a(n)=(11*n+6)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(11*n+6)^2 for n in (0..50)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (36 +181*x +25*x^2)/(1-x)^3.
E.g.f.: (36 +253*x +121*x^2)*exp(x). (End)

A017463 a(n) = (11*n + 6)^3.

Original entry on oeis.org

216, 4913, 21952, 59319, 125000, 226981, 373248, 571787, 830584, 1157625, 1560896, 2048383, 2628072, 3307949, 4096000, 5000211, 6028568, 7189057, 8489664, 9938375, 11543176, 13312053, 15252992, 17373979, 19683000, 22188041, 24897088, 27818127, 30959144

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+6)^m: A017461 (m=1), A017462 (m=2), this sequence (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..40], n-> (11*n+6)^3); # G. C. Greubel, Sep 19 2019

[(11*n+6)^3: n in [0..40]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+6)^3, n=0..40); # G. C. Greubel, Sep 19 2019

(* From Harvey P. Dale, May 16 2012 : (Start) *)
(11Range[0,40]+6)^3
LinearRecurrence[{4,-6,4,-1}, {216,4913, 21952,59319}, 40] (* End *)

vector(40, n, (11*n-5)^3) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^3 for n in (0..40)] # G. C. Greubel, Sep 19 2019

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=216, a(1)=4913, a(2)=21952, a(3)=59319. - Harvey P. Dale, May 16 2012
From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (216 +4049*x +3596*x^2 +125*x^3)/(1-x)^4.
E.g.f.: (216 +4697*x +6171*x^2 +1331*x^3)*exp(x). (End)

A017464 a(n) = (11*n + 6)^4.

Original entry on oeis.org

1296, 83521, 614656, 2313441, 6250000, 13845841, 26873856, 47458321, 78074896, 121550625, 181063936, 260144641, 362673936, 492884401, 655360000, 855036081, 1097199376, 1387488001, 1731891456, 2136750625, 2608757776, 3154956561, 3782742016, 4499860561, 5314410000

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), this sequence (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..30], n-> (11*n+6)^4); # G. C. Greubel, Sep 19 2019

[(11*n+6)^4: n in [0..30]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+6)^4, n=0..30); # G. C. Greubel, Sep 19 2019

(11*Range[30] -5)^4 (* G. C. Greubel, Sep 19 2019 *)
LinearRecurrence[{5,-10,10,-5,1},{1296,83521,614656,2313441,6250000},30] (* Harvey P. Dale, Oct 11 2021 *)

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

[(11*n+5)^4 for n in (0..30)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (1296 +77041*x +210011*x^2 +62411*x^3 +625*x^4)/(1-x)^5.
E.g.f.: (1296 +82225*x +224455*x^2 +119790*x^3 +14641*x^4)*exp(x). (End)

A017465 a(n) = (11*n + 6)^5.

Original entry on oeis.org

7776, 1419857, 17210368, 90224199, 312500000, 844596301, 1934917632, 3939040643, 7339040224, 12762815625, 21003416576, 33038369407, 50049003168, 73439775749, 104857600000, 146211169851, 199690286432, 267785184193, 353305857024, 459401384375, 589579257376

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), this sequence (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..30], n-> (11*n+6)^5); # G. C. Greubel, Sep 19 2019

[(11*n+6)^5: n in [0..30]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+6)^5, n=0..30); # G. C. Greubel, Sep 19 2019

(11*Range[30] -5)^5 (* G. C. Greubel, Sep 19 2019 *)

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

[(11*n+6)^5 for n in (0..30)] # G. C. Greubel, Sep 19 2019

G.f.: (7776 +1373201*x +8807866*x^2 +8104326*x^3 +1029826*x^4 +3125*x^5 )/(1-x)^6. - Colin Barker, Sep 17 2012

E.g.f.: (7776 +1412081*x +7189215*x^2 +7140815*x^3 +2049740*x^4 + 161051*x^5)*exp(x). - G. C. Greubel, Sep 19 2019

A017466 a(n) = (11*n + 6)^6.

Original entry on oeis.org

46656, 24137569, 481890304, 3518743761, 15625000000, 51520374361, 139314069504, 326940373369, 689869781056, 1340095640625, 2436396322816, 4195872914689, 6906762437184, 10942526586601, 16777216000000, 25002110044521, 36343632130624, 51682540549249, 72074394832896

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), this sequence (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

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

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

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

(11 * Range[0, 20] + 6)^6 (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {46656, 24137569, 481890304, 3518743761, 15625000000, 51520374361, 139314069504}, 20] (* Harvey P. Dale, Jan 19 2013 *)

a(n)=(11*n+6)^6 \\ Charles R Greathouse IV, Nov 04 2017

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

a(0) = 46656, a(1) = 24137569, a(2) = 481890304, a(3) = 3518743761, a(4) = 15625000000, a(5) = 51520374361, a(6) = 139314069504, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Jan 19 2013
From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (46656 +23810977*x +313907097*x^2 +650767622*x^3 +270308102*x^4 +16667841*x^5 +15625*x^6)/(1-x)^7.
E.g.f.: (46656 +24090913*x +216830911*x^2 +357573150*x^3 +181035965*x^4 +32371251*x^5 +1771561*x^6)*exp(x). (End)

A017467 a(n) = (11*n + 6)^7.

Original entry on oeis.org

279936, 410338673, 13492928512, 137231006679, 781250000000, 3142742836021, 10030613004288, 27136050989627, 64847759419264, 140710042265625, 282621973446656, 532875860165503, 953133216331392, 1630436461403549, 2684354560000000, 4275360817613091, 6614541047773568

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), this sequence (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

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

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

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

(11 Range[0, 19] + 6)^7 (* Alonso del Arte, Nov 04 2017 *)

vector(20, n, (11*n-5)^7) \\ G. C. Greubel, Sep 19 2019

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

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (279936 +408099185*x +10218057336*x^2 +40761385011*x^3 +38244574736 *x^4 +8315057055*x^5 +267810456*x^6 +78125*x^7)/(1-x)^8.
E.g.f.: (279936 +410058737*x +6336265551*x^2 +16330492871*x^3 + 12985102900*x^4 +3966041926*x^5 +483636153*x^6 +19487171*x^7)*exp(x). (End)

A017468 a(n) = (11*n + 6)^8.

Original entry on oeis.org

1679616, 6975757441, 377801998336, 5352009260481, 39062500000000, 191707312997281, 722204136308736, 2252292232139041, 6095689385410816, 14774554437890625, 32784148919812096, 67675234241018881

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), this sequence (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..20], n-> (11*n+6)^8); # G. C. Greubel, Sep 19 2019

[(11*n+6)^8: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+6)^8, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[20] -5)^8 (* G. C. Greubel, Sep 19 2019 *)

vector(20, n, (11*n-5)^8) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^8 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (1679616 +6960640897*x +315080647543*x^2 +2202777455589*x^3 + 3909536602339*x^4 +1960512320323*x^5 +243788893317*x^6 +4291451671*x^7 + 390625*x^8)/(1-x)^9.
E.g.f.: (1679616 +6974077825*x +181926081535*x^2 +706588143030*x^3 + 828890566581*x^4 +384764689386*x^5 +78448264202*x^6 +6937432876*x^7 + 214358881*x^8)*exp(x). (End)

A017469 a(n) = (11*n + 6)^9.

Original entry on oeis.org

10077696, 118587876497, 10578455953408, 208728361158759, 1953125000000000, 11694146092834141, 51998697814228992, 186940255267540403, 572994802228616704, 1551328215978515625, 3802961274698203136

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), this sequence (m=9), A017470 (m=10), A017471 (m=11), A017472 (m=12).

List([0..20], n-> (11*n+6)^9); # G. C. Greubel, Sep 19 2019

[(11*n+6)^9: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+6)^9, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[0,20]+6)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210, 120,-45,10,-1}, {10077696, 118587876497,10578455953408, 208728361158759, 1953125000000000,11694146092834141,51998697814228992,186940255267540403, 572994802228616704, 1551328215978515625}, 20] (* Harvey P. Dale, Jan 15 2019 *)

vector(20, n, (11*n-5)^9) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^9 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (10077696 +118487099537*x +9393030684758*x^2 +108279046743524*x^3 + 327643477452290*x^4 +311158545054314*x^5 +92052268491098*x^6 + 6938490608252*x^7 +68699945486*x^8 +1953125*x^9)/(1-x)^10.
E.g.f.: (10077696 +118577798801*x +5170645139055*x^2 +29558124475055*x^3 +49216997902380*x^4 +32588442284937*x^5 +9880686605790*x^6 + 1438737834930*x^7 +96461496450*x^8 +2357947691*x^9)*exp(x). (End)

A017470 a(n) = (11*n + 6)^10.

Original entry on oeis.org

60466176, 2015993900449, 296196766695424, 8140406085191601, 97656250000000000, 713342911662882601, 3743906242624487424, 15516041187205853449, 53861511409489970176, 162889462677744140625

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+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), this sequence (m=10), A017471 (m=11), A017472 (m=12).

List([0..20], n-> (11*n+6)^10); # G. C. Greubel, Sep 19 2019

[(11*n+6)^10: n in [0..10]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+6)^10, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[20] -5)^10 (* G. C. Greubel, Sep 19 2019 *)

vector(20, n, (11*n-5)^10) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^10 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019:(Start)
G.f.: (60466176 +2015328772513*x +274024159430165*x^2 +4993111339147592* x^3 +24069986191404704*x^4 +38639279895450554*x^5 +21874532039020442*x^6 +4073880923146640*x^7 +193797041298488*x^8 +1099404205901*x^9 +9765625* x^10)/(1-x)^11.
E.g.f.: (60466176 +2015933434273*x +146082419680351*x^2 +1209643951056750 *x^3 +2785989264344605*x^4 +2529281956307337*x^5 +1069882300751187*x^6 + 228102792962880*x^7 +24893496850530*x^8 +1308660968505*x^9 +25937424601* x^10)*exp(x). (End)

cp's OEIS Frontend

A269044 a(n) = 13*n + 7.

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A017462 a(n) = (11*n + 6)^2.

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A017463 a(n) = (11*n + 6)^3.

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A017464 a(n) = (11*n + 6)^4.

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Sage

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A017465 a(n) = (11*n + 6)^5.

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GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A017466 a(n) = (11*n + 6)^6.

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Links

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GAP

Magma