A105636 -id:A105636 - cp's OEIS frontend

A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1.

0, 1, 16, 82, 272, 707, 1568, 3108, 5664, 9669, 15664, 24310, 36400, 52871, 74816, 103496, 140352, 187017, 245328, 317338, 405328, 511819, 639584, 791660, 971360, 1182285, 1428336, 1713726, 2042992, 2421007, 2852992, 3344528, 3901568, 4530449, 5237904

Offset: 0

Stanislav Sykora, Nov 07 2013

In physics, a(n)/2^(M-1) is the trace of the spin operator |S_z|^M for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of their 4th powers is 2*82/16 = a(3)/8 (analogously for other values of M).

Partial sums of A062392. - Bruce J. Nicholson, Jun 29 2019

			a(4) = 4^4 + 2^4 = 272; a(5) = 5^4 + 3^4 + 1^4 = 707.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

Cf. A000389, A000579, A000332, A000583.

List([0..40], n-> n*(3*n^4+15*n^3+20*n^2-8)/30); # G. C. Greubel, Jul 01 2019

[1/30*n*(3*n^4+15*n^3+20*n^2-8): n in [0..40]]; // Vincenzo Librandi, Dec 23 2015

Table[SeriesCoefficient[x*(1+10*x+x^2)/(1-x)^6, {x, 0, n}], {n, 0, 40}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 16, 82, 272, 707}, 40] (* Vincenzo Librandi, Dec 23 2015 *)

nmax=40;a = vector(nmax);a[2]=1;for(i=3,#a,a[i]=a[i-2]+(i-1)^4); print(a);

concat(0, Vec(x*(1+10*x+x^2)/(1-x)^6 + O(x^40))) \\ Colin Barker, Dec 22 2015

[n*(3*n^4+15*n^3+20*n^2-8)/30 for n in (0..40)] # G. C. Greubel, Jul 01 2019

a(n) = Sum_{k=0..floor(n/2)} (n - 2*k)^4.
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/30)*n*(3*n^4 + 15*n^3 + 20*n^2 - 8).
G.f.: x*(1 + 10*x + x^2) / (1-x)^6.
(End)
E.g.f.: x*(30 + 210*x + 185*x^2 + 45*x^3 + 3*x^4)*exp(x)/30. - G. C. Greubel, Apr 24 2016
From Bruce J. Nicholson, Jun 29 2019: (Start)
a(n) = 12*A000389(n+3) + A000292(n);
a(n) = (12*A000579(n+4)+A000332(n+3)) - (12*A000579(n+3)+A000332(n+2));
a(n) - a(n-2) = A000583(n). (End)

A231304 Recurrence a(n) = a(n-2) + n^M for M=5, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 32, 244, 1056, 3369, 8832, 20176, 41600, 79225, 141600, 240276, 390432, 611569, 928256, 1370944, 1976832, 2790801, 3866400, 5266900, 7066400, 9351001, 12220032, 15787344, 20182656, 25552969, 32064032, 39901876, 49274400, 60413025, 73574400, 89042176

Offset: 0

Stanislav Sykora, Nov 07 2013

See A231303.

			a(4) = 4^5 + 2^5 = 1056; a(5) = 5^5 + 3^5 + 1^5 = 3369.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+n^5},a,{n,30}] (* or *) LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{0,1,32,244,1056,3369,8832,20176},40] (* Harvey P. Dale, Jul 22 2014 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^5); print(a);

concat(0, Vec(x*(1+26*x+66*x^2+26*x^3+x^4)/((1-x)^7*(1+x)) + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^5.
a(0)=0, a(1)=1, a(2)=32, a(3)=244, a(4)=1056, a(5)=3369, a(6)=8832, a(7)=20176, a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8). - Harvey P. Dale, Jul 22 2014
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/24)*(2*n^6 + 12*n^5 + 20*n^4 - 16*n^2 - 3*(-1)^n + 3).
G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4) / ((1-x)^7*(1+x)).
(End)

A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 64, 730, 4160, 16355, 50816, 134004, 312960, 665445, 1312960, 2437006, 4298944, 7263815, 11828480, 18654440, 28605696, 42792009, 62617920, 89837890, 126617920, 175604011, 239997824, 323639900, 431100800, 567780525, 740016576, 955201014, 1221906880

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^6 + 3^6 + 1^6 = 16355.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

map(op,ListTools:-PartialSums([seq([(2*i)^6,(2*i+1)^6],i=0..50)])); # Robert Israel, Dec 22 2015

Table[SeriesCoefficient[x (1 + 56 x + 246 x^2 + 56 x^3 + x^4)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,64,730,4160,16355,50816,134004},30] (* Harvey P. Dale, Aug 23 2025 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^6); print(a);

concat(0, Vec(x*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^8 + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)}(n-2k)^6.
From Colin Barker, Dec 22 2015: (Start)
a(n) = 1/42*n*(3*n^6+21*n^5+42*n^4-56*n^2+32).
G.f.: x*(1+56*x+246*x^2+56*x^3+x^4) / (1-x)^8.
(End)

A231306 Recurrence a(n) = a(n-2) + n^M for M=7, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 128, 2188, 16512, 80313, 296448, 903856, 2393600, 5686825, 12393600, 25173996, 48225408, 87922513, 153638912, 258781888, 422074368, 669120561, 1034294400, 1562992300, 2314294400, 3364080841, 4808652288, 6768906288, 9395123712, 12872421913, 17426933888

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^7 + 3^7 + 1^7 = 80313.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-27,48,-42,0,42,-48,27,-8,1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231307 (M=8), A231308 (M=9), A231309 (M=10).

Table[SeriesCoefficient[x (1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/((1 - x)^9 (1 + x)), {x, 0, n}], {n, 0, 26}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{8, -27, 48, -42, 0, 42, -48, 27, -8, 1}, {0, 1, 128, 2188, 16512, 80313, 296448, 903856, 2393600, 5686825}, 30] (* Vincenzo Librandi, Dec 23 2015 *)
nxt[{n_,a_,b_}]:={n+1,b,a+(n+1)^7}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Jun 16 2022 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^7); print(a);

concat(0, Vec(x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/((1-x)^9*(1+x)) + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^7.
From Colin Barker, Dec 22 2015: (Start)
a(n) = 1/96*(6*n^8+48*n^7+112*n^6-224*n^4+256*n^2+51*((-1)^n-1)).
G.f.: x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6) / ((1-x)^9*(1+x)).
(End)

A231307 Recurrence a(n) = a(n-2) + n^M for M=8, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 256, 6562, 65792, 397187, 1745408, 6161988, 18522624, 49208709, 118522624, 263567590, 548504320, 1079298311, 2024293376, 3642188936, 6319260672, 10617946377, 17339221248, 27601509418, 42939221248, 65424368779, 97815094784, 143735354060

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^8 + 3^8 + 1^8 = 397187.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231308 (M=9), A231309 (M=10).

[1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192): n in [0..30]]; // Vincenzo Librandi, Dec 23 2015

Table[1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192),{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^8); print(a);

concat(0, Vec(x*(1+246*x+4047*x^2+11572*x^3+4047*x^4+246*x^5+x^6)/(1-x)^10 + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^8.
a(n) = 1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192). - Vaclav Kotesovec, Feb 14 2014
G.f.: x*(1+246*x+4047*x^2+11572*x^3+4047*x^4+246*x^5+x^6) / (1-x)^10.

A231308 Recurrence a(n) = a(n-2) + n^M for M=9, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 512, 19684, 262656, 1972809, 10340352, 42326416, 144558080, 429746905, 1144558080, 2787694596, 6304338432, 13392193969, 26965385216, 51835553344, 95684861952, 170423429841, 294044152320, 493111127620, 806044152320, 1287391174201, 2013313370112

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^9 + 3^9 + 1^9 = 1972809.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-44,110,-165,132,0,-132,165,-110,44,-10,1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231309 (M=10).

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+n^9},a,{n,30}] (* or *)
LinearRecurrence[{10,-44,110,-165,132,0,-132,165,-110,44,-10,1},{0,1,512,19684,262656,1972809,10340352,42326416,144558080,429746905,1144558080,2787694596},30] (* Harvey P. Dale, Apr 29 2014 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^9); print(a);

concat(0, Vec(x*(1 +502*x +14608*x^2 +88234*x^3 +156190*x^4 +88234*x^5 +14608*x^6 +502*x^7 +x^8) / ((1 -x)^11*(1 +x)) + O(x^40))) \\ Colin Barker, Dec 22 2015

a(n) = Sum{k=0..floor(n/2)}(n-2k)^9.
a(0)=0, a(1)=1, a(2)=512, a(3)=19684, a(4)=262656, a(5)=1972809, a(6)=10340352, a(7)=42326416, a(8)=144558080, a(9)=429746905, a(10)=1144558080, a(11)=2787694596, a(n) = 10*a(n-1) - 44*a(n-2) + 110*a(n-3) - 165*a(n-4) + 132*a(n-5) - 132*a(n-7) + 165*a(n-8) - 110*a(n-9) + 44*a(n-10) - 10*a(n-11) + a(n-12). - Harvey P. Dale, Apr 29 2014
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/40)*(2*n^10 + 20*n^9 + 60*n^8 - 224*n^6 + 640*n^4 - 768*n^2 - 155*((-1)^n -1)).
G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8) / ((1-x)^11*(1+x)). (End)

PARI code corrected by Colin Barker, Dec 22 2015

A231309 Recurrence a(n) = a(n-2) + n^M for M=10, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1024, 59050, 1049600, 9824675, 61515776, 292299924, 1135257600, 3779084325, 11135257600, 29716508926, 73052621824, 167575000775, 362307276800, 744225391400, 1461818904576, 2760219291849, 5032286131200, 8891285549650, 15272286131200

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^10 + 3^10 + 1^10 = 9824675.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9).

CoefficientList[Series[x (1 + 1012 x + 46828 x^2 + 408364 x^3 + 901990 x^4 + 408364 x^5 + 46828 x^6 + 1012 x^7 + x^8)/(1 - x)^12, {x, 0, 20}], x] (* Michael De Vlieger, Nov 20 2019 *)

nmax=100; a=vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^10); print(a);

concat(0, Vec(x*(1 +1012*x +46828*x^2 +408364*x^3 +901990*x^4 +408364*x^5 +46828*x^6 +1012*x^7 +x^8) / (1 -x)^12 + O(x^40))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n\2)} (n-2*k)^10.
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/66)*n*(3*n^10 + 33*n^9 + 110*n^8 - 528*n^6 + 2112*n^4 - 4224*n^2 + 2560).
G.f.: x*(1 + 1012*x + 46828*x^2 + 408364*x^3 + 901990*x^4 + 408364*x^5 + 46828*x^6 + 1012*x^7 + x^8) / (1-x)^12.
(End)

A254371 Sum of cubes of the first n even numbers (A016743).

Original entry on oeis.org

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Luciano Ancora, Mar 16 2015

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
Diophante, Une miniature avec trois entiers consécutifs (in French).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
Cf. A060300 (2*a(n)).
First bisection of A105636; second bisection of A212892.
Cf. A000217, A000415, A002378, A035287, A163102.

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
a(n) = 2*n^2*(n + 1)^2.
a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017
a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*A163102(n) = A060300(n)/2. (End)

A289706 Number of 5-cycles in the n-triangular honeycomb queen graph.

Original entry on oeis.org

0, 0, 24, 324, 1692, 5796, 15516, 35388, 71988, 134460, 234972, 389304, 617400, 943992, 1399272, 2019528, 2847960, 3935304, 5340816, 7132860, 9390084, 12201948, 15670116, 19908900, 25046892, 31227300, 38609844, 47370960, 57705984, 69829200, 83976336, 100404432

Offset: 1

Eric W. Weisstein, Jul 14 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Cycle
Index entries for linear recurrences with constant coefficients, signature (4, -3, -8, 14, 0, -14, 8, 3, -4, 1).

Cf. A105636 (3-cycles), A289705 (4-cycles), A289707 (6-cycles).

Table[(1095 - 4770 n + 934 n^2 + 4680 n^3 - 3170 n^4 + 540 n^5 +
    16 n^6 - 15 (-1)^n (73 - 30 n + 2 n^2))/320, {n, 20}]
LinearRecurrence[{4, -3, -8, 14, 0, -14, 8, 3, -4, 1}, {0, 0, 24, 324, 1692, 5796, 15516, 35388, 71988, 134460}, 20]

concat(vector(2), Vec(12*x^3*(2 + 19*x + 39*x^2 + 16*x^3 - 28*x^4 - 24*x^5) / ((1 - x)^7*(1 + x)^3) + O(x^50))) \\ Colin Barker, Aug 07 2017

a(n) = (1095 - 4770*n + 934*n^2 + 4680*n^3 - 3170*n^4 + 540*n^5 +
    16*n^6 - 15*(-1)^n (73 - 30*n + 2*n^2))/320.
a(n) = 4*a(n-1) - 3*a(n-2) - 8*a(n-3) + 14*a(n-4) - 14*a(n-6) + 8*a(n-7) + 3*a(n-8) - 4*a(n-9) + a(n-10).
G.f.: 12*x^3*(2 + 19*x + 39*x^2 + 16*x^3 - 28*x^4 - 24*x^5) / ((1 - x)^7*(1 + x)^3). - Colin Barker, Aug 07 2017

A289707 Number of 6-cycles in the n-triangular honeycomb queen graph.

Original entry on oeis.org

0, 0, 16, 911, 8013, 38130, 129932, 358272, 851710, 1815124, 3554910, 6510729, 11289019, 18704640, 29823436, 46014402, 69002190, 100930284, 144424446, 202667301, 279473821, 379377584, 507719550, 670746120, 875712560, 1130992902, 1446199474, 1832304547, 2301777585, 2868718404

Offset: 1

Eric W. Weisstein, Jul 14 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Cycle
Index entries for linear recurrences with constant coefficients, signature (3, 1, -10, 3, 13, -3, -12, -3, 13, 3, -10, 1, 3, -1).

Cf. A105636 (3-cycles), A289705 (4-cycles), A289706 (5-cycles).

Table[(-315 (-1)^n (-2489 + 1659 n - 297 n^2 + 10 n^3) - 792995 + 3789081 n - 1968939 n^2 - 3033450 n^3 + 3489990 n^4 - 1269366 n^5 + 154014 n^6 + 1440 n^7 + 8960 Cos[2 n Pi/3] - 8960 Sqrt[3] Sin[2 n Pi/3])/40320, {n, 20}]
LinearRecurrence[{3, 1, -10, 3, 13, -3, -12, -3, 13, 3, -10, 1, 3, -1}, {0, 0, 16, 911, 8013, 38130, 129932, 358272, 851710, 1815124, 3554910, 6510729, 11289019, 18704640}, 20]

concat(vector(2), Vec(x^3*(16 + 863*x + 5264*x^2 + 13340*x^3 + 16591*x^4 + 7535*x^5 - 7572*x^6 - 14592*x^7 - 9919*x^8 - 2886*x^9) / ((1 - x)^8*(1 + x)^4*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Jul 27 2017

a(n) = 3*a(n-1)+a(n-2)-10*a(n-3)+3*a(n-4)+13*a(n-5)-3*a(n-6)-12*a(n-7)-3*a(n-8)+13*a(n-9)+3*a(n-10)-10*a(n-11)+a(n-12)+3*a(n-13)-a(n-14).

G.f.: x^3*(16 + 863*x + 5264*x^2 + 13340*x^3 + 16591*x^4 + 7535*x^5 - 7572*x^6 - 14592*x^7 - 9919*x^8 - 2886*x^9) / ((1 - x)^8*(1 + x)^4*(1 + x + x^2)). - Colin Barker, Jul 27 2017

cp's OEIS Frontend

A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1.

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A231304 Recurrence a(n) = a(n-2) + n^M for M=5, starting with a(0)=0, a(1)=1.

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A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.

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A231306 Recurrence a(n) = a(n-2) + n^M for M=7, starting with a(0)=0, a(1)=1.

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A231307 Recurrence a(n) = a(n-2) + n^M for M=8, starting with a(0)=0, a(1)=1.

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A231308 Recurrence a(n) = a(n-2) + n^M for M=9, starting with a(0)=0, a(1)=1.

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